A Brauer's theorem and related results

Given a square matrix A, a Brauer's theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75-91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt's and Hotelling's deflations. An extension of the aforementioned Brauer's result, Rado's theorem, shows how to modify r eigenvalues of A at the same time via a rank-r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem. © 2012 Versita Warsaw and Springer-Verlag Wien.This work is supported by Fondecyt 1085125, Chile, the Spanish grant DGI MTM2010-18228 and the Programa de Apoyo a la Investigacion y Desarrollo (PAID-06-10) of the UPV.Bru García, R.; Cantó Colomina, R.; Soto, RL.; Urbano Salvador, AM. (2012). A Brauer's theorem and related results. Central European Journal of Mathematics. 10(1):312-321. https://doi.org/10.2478/s11533-011-0113-0S312321101Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75–91Crouch P.E., Introduction to Mathematical Systems Theory, Mathematik-Arbeitspapiere, Bremen, 1988Delchamps D.F., State-Space and Input-Output Linear Systems, Springer, New York, 1988Hautus M.L.J., Controllability and observability condition of linear autonomous systems, Nederl. Akad. Wetensch. Indag. Math., 1969, 72, 443–448Kailath T., Linear Systems, Prentice Hall Inform. System Sci. Ser., Prentice Hall, Englewood Cliffs, 1980Langville A.N., Meyer C.D., Deeper inside PageRank, Internet Math., 2004, 1(3), 335–380Perfect H., Methods of constructing certain stochastic matrices. II, Duke Math. J., 1955, 22(2), 305–311Saad Y., Numerical Methods for Large Eigenvalue Problems, Classics Appl. Math., 66, SIAM, Philadelphia, 2011Soto R.L., Rojo O., Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem, Linear Algebra Appl., 2006, 416(2–3), 844–856Wilkinson J.H., The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 196

A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

On the Influence of Nb/Ti Ratio on Environmentally-Assisted Crack Growth in High-Strength Nickel-Based Superalloys

The effect of Nb/Ti ratio on environmentally-assisted crack growth of three prototype Ni-based superalloys is studied. For these alloys, the yield strength is unaltered with increasing Nb/Ti ratio due to an increase in grain size. This situation has allowed the rationalization of the factors influencing damage tolerance at 700 °C. Primary intergranular cracks have been investigated using energy-dispersive X-ray spectroscopy in a scanning transmission electron microscope and the analysis of electron back-scatter diffraction patterns. Any possible detrimental effect of Nb on the observed crack tip damage due to Nb-rich oxide formation is not observed. Instead, evidence is presented to indicate that the tertiary γ′-precipitates are dissolving ahead of the crack consistent with the formation of oxides such as alumina and rutile. Our results have implications for alloy design efforts; at any given strength level, both more and less damage-tolerant variants of these alloys can be designed

Search for charginos in e+e- interactions at sqrt(s) = 189 GeV

An update of the searches for charginos and gravitinos is presented, based on a data sample corresponding to the 158 pb^{-1} recorded by the DELPHI detector in 1998, at a centre-of-mass energy of 189 GeV. No evidence for a signal was found. The lower mass limits are 4-5 GeV/c^2 higher than those obtained at a centre-of-mass energy of 183 GeV. The (\mu,M_2) MSSM domain excluded by combining the chargino searches with neutralino searches at the Z resonance implies a limit on the mass of the lightest neutralino which, for a heavy sneutrino, is constrained to be above 31.0 GeV/c^2 for tan(beta) \geq 1.Comment: 22 pages, 8 figure

Search for composite and exotic fermions at LEP 2

A search for unstable heavy fermions with the DELPHI detector at LEP is reported. Sequential and non-canonical leptons, as well as excited leptons and quarks, are considered. The data analysed correspond to an integrated luminosity of about 48 pb^{-1} at an e^+e^- centre-of-mass energy of 183 GeV and about 20 pb^{-1} equally shared between the centre-of-mass energies of 172 GeV and 161 GeV. The search for pair-produced new leptons establishes 95% confidence level mass limits in the region between 70 GeV/c^2 and 90 GeV/c^2, depending on the channel. The search for singly produced excited leptons and quarks establishes upper limits on the ratio of the coupling of the excited fermio

Search for lightest neutralino and stau pair production in light gravitino scenarios with stau NLSP

Promptly decaying lightest neutralinos and long-lived staus are searched for in the context of light gravitino scenarios. It is assumed that the stau is the next to lightest supersymmetric particle (NLSP) and that the lightest neutralino is the next to NLSP (NNLSP). Data collected with the Delphi detector at centre-of-mass energies from 161 to 183 \GeV are analysed. No evidence of the production of these particles is found. Hence, lower mass limits for both kinds of particles are set at 95% C.L.. The mass of gaugino-like neutralinos is found to be greater than 71.5 GeV/c^2. In the search for long-lived stau, masses less than 70.0 to 77.5 \GeVcc are excluded for gravitino masses from 10 to 150 \eVcc . Combining this search with the searches for stable heavy leptons and Minimal Supersymmetric Standard Model staus a lower limit of 68.5 \GeVcc may be set for the stau mas

Hadronization properties of b quarks compared to light quarks in e+e- -> q qbar from 183 to 200 GeV

The DELPHI detector at LEP has collected 54 pb^{-1} of data at a centre-of-mass energy around 183 GeV during 1997, 158 pb^{-1} around 189 GeV during 1998, and 187 pb^{-1} between 192 and 200 GeV during 1999. These data were used to measure the average charged particle multiplicity in e+e- -> b bbar events, _{bb}, and the difference delta_{bl} between _{bb} and the multiplicity, _{ll}, in generic light quark (u,d,s) events: delta_{bl}(183 GeV) = 4.55 +/- 1.31 (stat) +/- 0.73 (syst) delta_{bl}(189 GeV) = 4.43 +/- 0.85 (stat) +/- 0.61 (syst) delta_{bl}(200 GeV) = 3.39 +/- 0.89 (stat) +/- 1.01 (syst). This result is consistent with QCD predictions, while it is inconsistent with calculations assuming that the multiplicity accompanying the decay of a heavy quark is independent of the mass of the quark itself.Comment: 13 pages, 2 figure

Updated Nucleosynthesis Constraints on Unstable Relic Particles

We revisit the upper limits on the abundance of unstable massive relic particles provided by the success of Big-Bang Nucleosynthesis calculations. We use the cosmic microwave background data to constrain the baryon-to-photon ratio, and incorporate an extensively updated compilation of cross sections into a new calculation of the network of reactions induced by electromagnetic showers that create and destroy the light elements deuterium, he3, he4, li6 and li7. We derive analytic approximations that complement and check the full numerical calculations. Considerations of the abundances of he4 and li6 exclude exceptional regions of parameter space that would otherwise have been permitted by deuterium alone. We illustrate our results by applying them to massive gravitinos. If they weigh ~100 GeV, their primordial abundance should have been below about 10^{-13} of the total entropy. This would imply an upper limit on the reheating temperature of a few times 10^7 GeV, which could be a potential difficulty for some models of inflation. We discuss possible ways of evading this problem.Comment: 40 pages LaTeX, 18 eps figure