Asymptotic properties of eigenmatrices of a large sample covariance matrix

Let $S_n=\frac{1}{n}X_nX_n^*$ where $X_n=\{X_{ij}\}$ is a $p\times n$ matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_n(\mathbf {t}_1,\mathbf {t}_2,\sigma)=\sqrt{p}({\mathbf {x}}_n(\mathbf {t}_1)^*(S_n+\sigma I)^{-1}{\mathbf {x}}_n(\mathbf {t}_2)-{\mathbf {x}}_n(\mathbf {t}_1)^*{\mathbf {x}}_n(\mathbf {t}_2)m_n(\sigma))$ in which $\sigma>0$ and $m_n(\sigma)=\int\frac{dF_{y_n}(x)}{x+\sigma}$ where $F_{y_n}(x)$ is the Mar\v{c}enko--Pastur law with parameter $y_n=p/n$; which converges to a positive constant as $n\to\infty$, and ${\mathbf {x}}_n(\mathbf {t}_1)$ and ${\mathbf {x}}_n(\mathbf {t}_2)$ are unit vectors in ${\Bbb{C}}^p$, having indices $\mathbf {t}_1$ and $\mathbf {t}_2$, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence $Y_n(\mathbf {t}_1,\mathbf {t}_2,\sigma)$ converges weakly to a $(2m+1)$-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of $S_n$ is asymptotically close to that of a Haar-distributed unitary matrix.Comment: Published in at http://dx.doi.org/10.1214/10-AAP748 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Periodicities in sunspot activity during solar cycle 23

The data of sunspot numbers, sunspot areas and solar flare index during cycle 23 are analyzed to investigate the intermediate-term periodicities. Power spectral analysis has been performed separately for the data of the whole disk, northern and southern hemispheres of the Sun. Several significant midrange periodicities ($\sim$175, 133, 113, 104, 84, 63 days) are detected in sunspot activity. Most of the periodicities in sunspot numbers generally agree with those of sunspot areas during the solar cycle 23. The study reveals that the periodic variations in the northern and southern hemispheres of the Sun show a kind of asymmetrical behavior. Periodicities of $\sim$175 days and $\sim$133 days are highly significant in the sunspot data of northern hemisphere showing consistency with the findings of Lean (1990) during solar cycles 12-21. On the other hand, southern hemisphere shows a strong periodicity of about 85 days in terms of sunspot activity. The analysis of solar flare index data of the same time interval does not show any significant peak. The different periodic behavior of sunspot and flare activity can be understood in the light of hypothesis proposed by Ballester et al. (2002), which suggests that during cycle 23, the periodic emergence of magnetic flux partly takes place away from developed sunspot groups and hence may not necessarily increase the magnetic complexity of sunspot groups that leads to the generation of flares.Comment: 4 pages, 4 figure

Impurity scattering and Friedel oscillations in mono-layer black phosphorus

We study the effect of impurity scattering effect in black phosphorurene (BP) in this work. For single impurity, we calculate impurity induced local density of states (LDOS) in momentum space numerically based on tight-binding Hamiltonian. In real space, we calculate LDOS and Friedel oscillation analytically. LDOS shows strong anisotropy in BP. Many impurities in BP are investigated using $T$-matrix approximation when the density is low. Midgap states appear in band gap with peaks in DOS. The peaks of midgap states are dependent on impurity potential. For finite positive potential, the impurity tends to bind negative charge carriers and vise versa. The infinite impurity potential problem is related to chiral symmetry in BP

Significantly enhanced critical current densities in MgB2 tapes made by a scaleable, nano-carbon addition route

Nanocarbon-doped Fe-sheathed MgB2 tapes with different doping levels were prepared by the in situ powder-in-tube method. Compared to the undoped tapes, Jc for all the C-doped samples was enhanced by more than an order of magnitude in magnetic fields above 9 T. At 4.2 K, the transport Jc for the 5 at% doped tapes reached 1.85x104 A/cm2 at 10 T and 2.8x103 A/cm2 at 14 T, respectively. Moreover, the critical temperature for the doped tapes decreased slightly. Transmission electron microscopy showed a number of intra-granular dislocations and the dispersed nanoparticles embedded within MgB2 grains induced by the C doping. The mechanism for the enhancement of flux pinning is also discussed. These results indicate that powder-in-tube-processed MgB2 tape is very promising for high-field applications.Comment: 13 pages, 5 figures. to be published soo

Adiabatic heating in impulsive solar flares

The dynamic X-ray spectra of two simple, impulsive solar flares are examined together with H alpha, microwave and meter wave radio observations. X-ray spectra of both events were characteristic of thermal bremsstrahlung from single temperature plasmas. The symmetry between rise and fall was found to hold for the temperature and emission measure. The relationship between temperature and emission measure was that of an adiabatic compression followed by adiabatic expansion; the adiabatic index of 5/3 indicated that the electron distribution remained isotropic. Observations in H alpha provided further evidence for compressive energy transfer