Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix

The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace \X, and a Hermitian matrix $A$, the eigenvalues of $X^HAX$ are called Ritz values of $A$ with respect to \X. If the subspace \X is $A$-invariant then the Ritz values are some of the eigenvalues of $A$. If the $A$-invariant subspace \X is perturbed to give rise to another subspace \Y, then the vector of absolute values of changes in Ritz values of $A$ represents the absolute eigenvalue approximation error using \Y. We bound the error in terms of principal angles between \X and \Y. We capitalize on ideas from a recent paper [DOI: 10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute values of differences between Ritz values for subspaces \X and \Y was weakly (sub-)majorized by a constant times the sine of the vector of principal angles between \X and \Y, the constant being the spread of the spectrum of $A$. In that result no assumption was made on either subspace being $A$-invariant. It was conjectured there that if one of the trial subspaces is $A$-invariant then an analogous weak majorization bound should only involve terms of the order of sine squared. Here we confirm this conjecture. Specifically we prove that the absolute eigenvalue error is weakly majorized by a constant times the sine squared of the vector of principal angles between the subspaces \X and \Y, where the constant is proportional to the spread of the spectrum of $A$. For many practical cases we show that the proportionality factor is simply one, and that this bound is sharp. For the general case we can only prove the result with a slightly larger constant, which we believe is artificial.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and Applications (SIMAX

On one mechanism of light ablation of nanostructures

The mechanism of mechanical ablation of nanoparticles during the interaction with a high-power laser radiation pulse is proposed. A particle is polarized under a laser electric field, and electric forces acting on field-induced oppositesign charges cause rupture stresses. Upon reaching the stresses exceeding the maximum allowable values for a given material, a nanoparticle decays into two ones. This effect can be used for narrowing the size distribution of nanoparticles produced by the laser ablation method

Lower bounds on the complexity of simulating quantum gates

We give a simple proof of a formula for the minimal time required to simulate a two-qubit unitary operation using a fixed two-qubit Hamiltonian together with fast local unitaries. We also note that a related lower bound holds for arbitrary n-qubit gates.Comment: 6 page

UV/Optical Detections of Candidate Tidal Disruption Events by GALEX and CFHTLS

We present two luminous UV/optical flares from the nuclei of apparently inactive early-type galaxies at z=0.37 and 0.33 that have the radiative properties of a flare from the tidal disruption of a star. In this paper we report the second candidate tidal disruption event discovery in the UV by the GALEX Deep Imaging Survey, and present simultaneous optical light curves from the CFHTLS Deep Imaging Survey for both UV flares. The first few months of the UV/optical light curves are well fitted with the canonical t^(-5/3) power-law decay predicted for emission from the fallback of debris from a tidally disrupted star. Chandra ACIS X-ray observations during the flares detect soft X-ray sources with T_bb= (2-5) x 10^5 K or Gamma > 3 and place limits on hard X-ray emission from an underlying AGN down to L_X (2-10 keV) <~ 10^41 ergs/s. Blackbody fits to the UV/optical spectral energy distributions of the flares indicate peak flare luminosities of > 10^44-10^45 ergs/s. The temperature, luminosity, and light curves of both flares are in excellent agreement with emission from a tidally disrupted main sequence star onto a central black hole of several times 10^7 msun. The observed detection rate of our search over ~ 2.9 deg^2 of GALEX Deep Imaging Survey data spanning from 2003 to 2007 is consistent with tidal disruption rates calculated from dynamical models, and we use these models to make predictions for the detection rates of the next generation of optical synoptic surveys.Comment: 28 pages, 27 figures, 11 tables, accepted to ApJ, final corrections from proofs adde

Discovery of an Ultrasoft X-ray Transient Source in the 2XMM Catalog: a Tidal Disruption Event Candidate

We have discovered an ultrasoft X-ray transient source, 2XMMi J184725.1-631724, which was detected serendipitously in two XMM-Newton observations in the direction of the center of the galaxy IC 4765-f01-1504 at a redshift of 0.0353. These two observations were separated by 211 days, with the 0.2-10 keV absorbed flux increasing by a factor of about 9. Their spectra are best described by a model dominated by a thermal disk or a single-temperature blackbody component (contributing >80% of the flux) plus a weak power-law component. The thermal emission has a temperature of a few tens of eV, and the weak power-law component has a photon index of ~3.5. Similar to the black hole X-ray binaries in the thermal state, our source exhibits an accretion disk whose luminosity appears to follow the $L\propto T^4$ relation. This would indicate that the black hole mass is about 10^5-10^6 M_sun using the best-fitting inner disk radius. Both XMM-Newton observations show variability of about 21% on timescales of hours, which can be explained as due to fast variations in the mass accretion rate. The source was not detected by ROSAT in an observation in 1992, indicating a variability factor of >64 over longer timescales. The source was not detected again in X-rays in a Swift observation in 2011 February, implying a flux decrease by a factor of >12 since the last XMM-Newton observation. The transient nature, in addition to the extreme softness of the X-ray spectra and the inactivity of the galaxy implied by the lack of strong optical emission lines, makes it a candidate tidal disruption event. If this is the case, the first XMM-Newton observation would have been in the rising phase, and the second one in the decay phase.Comment: 12 pages, 6 figures. Accepted for publication in Ap

Evolution of tidal disruption candidates discovered by XMM-Newton

It has been demonstrated that active galactic nuclei are powered by gas accretion onto supermassive black holes located at their centres. The paradigm that the nuclei of inactive galaxies are also occupied by black holes was predicted long ago by theory. In the last decade, this conjecture was confirmed by the discovery of giant-amplitude, non-recurrent X-ray flares from such inactive galaxies and explained in terms of outburst radiation from stars tidally disrupted by a dormant supermassive black hole at the nuclei of those galaxies. Due to the scarcity of detected tidal disruption events, the confirmation and follow-up of each new candidate is needed to strengthen the theory through observational data, as well as to shed new light on the characteristics of this type of events. Two tidal disruption candidates have been detected with XMM-Newton during slew observations. Optical and X-ray follow-up, post-outburst observations were performed on these highly variable objects in order to further study their classification and temporal evolution. We show that the detected low-state X-ray emission for these two candidates has properties such that it must still be related to the flare. The X-ray luminosity of the objects decreases according to theoretical predictions for tidal disruption events. At present, optical spectra of the sources do not present any evident signature of the disruption event. In addition, the tidal disruption rate as derived from the XMM-Newton slew survey has been computed and agrees with previous studies.Comment: 12 pages, 8 figures, A&A accepte

Principles of prediction of information protection in the communication network

An active method for combating unauthorized intrusion into the communication network operation is discussed. Preventive measures are planned based on known data on vulnerability of software products. An open NVD database is used. The most dangerous threats are predicted in the form of a game between two partners: the attacker and the defender. The game results in recommendations on information protection for the studied program system. © 2012 by Begell House, Inc