Moments of a single entry of circular orthogonal ensembles and Weingarten calculus

Consider a symmetric unitary random matrix $V=(v_{ij})_{1 \le i,j \le N}$ from a circular orthogonal ensemble. In this paper, we study moments of a single entry $v_{ij}$. For a diagonal entry $v_{ii}$ we give the explicit values of the moments, and for an off-diagonal entry $v_{ij}$ we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size $N$. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.Comment: 17 page

ASCA observations of massive medium-distant clusters of galaxies. II

We have selected seven medium-distant clusters of galaxies (z ~ 0.1 - 0.3) for multi-wavelength observations with the goal of investigating their dynamical state. Following Paper I (Pierre et al. 1999) which reported the ASCA results about two of them, we present here the analysis of the ASCA observations of the other five clusters; RXJ1023.8-2715 (A3444), RXJ1031.6-2607, RXJ1050.5-0236 (A1111), RXJ1203.2-2131(A1451), and RXJ1314.5-2517. Except for RXJ1031.6, whose X-ray emission turned out to be dominated by an AGN, the ASCA spectra are well fitted by a one-temperature thin thermal plasma model. We compare the temperature-luminosity relation of our clusters with that of nearby ones (z<0.1). Two clusters, RXJ1050.5 and RXJ1023.8, show larger luminosities than the bulk of clusters at similar temperatures, which suggests the presence of a cooling flow. The temperature vs. iron-abundance relationship of our sample is consistent with that of nearby clusters.Comment: 9 pages, 20 figures, A&AS in pres

Ac Susceptibility and Static Magnetization Measurements of CeRu2_2Si2_2 at Small Magnetic Fields and Ultra Low Temperatures

The magnetic properties of CeRu$_2$Si$_2$ at microkelvin temperatures (down to 170 $\mu$K) and ultra small magnetic fields ($0.02\sim6.21$ mT) are investigated experimentally for the first time. The simultaneously measured ac susceptibility and static magnetization show neither evidence of the magnetic ordering, superconductivity down to the lowest temperatures nor conventional Landau Fermi-Liquid behavior. The results imply the magnetic transition temperature in undoped CeRu$_2$Si$_2$ is very close to absolute 0 K. The possibility for proximity of CeRu$_2$Si$_2$ to the quantum critical point without any doping is discussed.Comment: 4 pages, 3 figures; accepted for publication in Phys. Rev. B (Rapid Communication) and scheduled issue on 1st of May 200

Lower bound for the quantum capacity of a discrete memoryless quantum channel

We generalize the random coding argument of stabilizer codes and derive a lower bound on the quantum capacity of an arbitrary discrete memoryless quantum channel. For the depolarizing channel, our lower bound coincides with that obtained by Bennett et al. We also slightly improve the quantum Gilbert-Varshamov bound for general stabilizer codes, and establish an analogue of the quantum Gilbert-Varshamov bound for linear stabilizer codes. Our proof is restricted to the binary quantum channels, but its extension of to l-adic channels is straightforward.Comment: 16 pages, REVTeX4. To appear in J. Math. Phys. A critical error in fidelity calculation was corrected by using Hamada's result (quant-ph/0112103). In the third version, we simplified formula and derivation of the lower bound by proving p(Gamma)+q(Gamma)=1. In the second version, we added an analogue of the quantum Gilbert-Varshamov bound for linear stabilizer code

NMR C-NOT gate through Aharanov-Anandan's phase shift

Recently, it is proposed to do quantum computation through the Berry's phase(adiabatic cyclic geometric phase) shift with NMR (Jones et al, Nature, 403, 869(2000)). This geometric quantum gate is hopefully to be fault tolerant to certain types of errors because of the geometric property of the Berry phase. Here we give a scheme to realize the NMR C-NOT gate through Aharonov-Anandan's phase(non-adiabatic cyclic phase) shift on the dynamic phase free evolution loop. In our scheme, the gate is run non-adiabatically, thus it is less affected by the decoherence. And, in the scheme we have chosen the the zero dynamic phase time evolution loop in obtaining the gepmetric phase shift, we need not take any extra operation to cancel the dynamic phase.Comment: 5 pages, 1 figur

Energy landscape and phase transitions in the self-gravitating ring model

We apply a recently proposed criterion for the existence of phase transitions, which is based on the properties of the saddles of the energy landscape, to a simplified model of a system with gravitational interactions, referred to as the self-gravitating ring model. We show analytically that the criterion correctly singles out the phase transition between a homogeneous and a clustered phase and also suggests the presence of another phase transition, not previously known. On the basis of the properties of the energy landscape we conjecture on the nature of the latter transition

HST-FOS Observations of M87: Ly-a Emission from the Active Galactic Nucleus

The Faint Object Spectrograph on the Hubble Space Telescope was used to obtain spectra of the central region of M87. These spectra cover the wavelength range 1140 - 1606 Angstrom and have a resolution of about 1 Angstrom. The nuclear continuum is clearly visible in the spectra. The only strong line that is observed is Ly-alpha, which has a velocity width of about 3000 km/sec. There is also a marginal detection of C IV 1549. The ratio of Ly-alpha to C IV in the nuclear spectrum is at least a factor of 2 higher than in a spectrum taken at a position on the disk about 0.6" away from the nucleus by Dopita et al. This enhancement of Ly-alpha at the nucleus could point to significant differences in the properties of the emitting gas and/or the excitation mechanism between the outer and inner disk regions. The strength of the observed Ly-alpha places limits on the properties of the absorbing gas present within M87. For instance, if the hydrogen column at the systemic velocity of M87 is greater than about 10^18 cm^{-2} then it can cover only a small fraction of the line emitting region. Spectra separated by 5 days show a 60% difference in the Ly-alpha flux, but the same continuum level. This could be due to either a displacement between the aperture positions for the two sets of observations, or it could be due to intrinsic variability of the source. The current observations do not strongly favor either of these alternatives. The observations do show, however, that the Ly-alpha line is a useful tracer of kinematics in the M87 nucleus.Comment: 14 pages + 5 figures. LaTeX uses aaspp4.sty. Accepted for publication in the Astrophysical Journa

General moments of the inverse real Wishart distribution and orthogonal Weingarten functions

Let $W$ be a random positive definite symmetric matrix distributed according to a real Wishart distribution and let $W^{-1}=(W^{ij})_{i,j}$ be its inverse matrix. We compute general moments $\mathbb{E} [W^{k_1 k_2} W^{k_3 k_4} ... W^{k_{2n-1}k_{2n}}]$ explicitly. To do so, we employ the orthogonal Weingarten function, which was recently introduced in the study for Haar-distributed orthogonal matrices. As applications, we give formulas for moments of traces of a Wishart matrix and its inverse.Comment: 29 pages. The last version differs from the published version, but it includes Appendi