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  • 2010-2014  (2)
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  • 1
    Publication Date: 2013-04-20
    Description: The regularity lemma of Szemerédi asserts that one can partition every graph into a bounded number of quasi-random bipartite graphs. In some applications however, one would like to have a strong control on how quasi-random these bipartite graphs are. Alon et al. (‘Efficient testing of large graphs’, Combinatorica 20 (2000) 451–476) obtained a powerful variant of the regularity lemma, which allows one to have an arbitrary control on this measure of quasi-randomness. However, their proof guaranteed only to produce a partition where the number of parts is given by the Wowzer function, which is the iterated version of the Tower function. We show here that a bound of this type is unavoidable by constructing a graph H , with the property that even if one wants a very mild control on the quasi-randomness of a regular partition, then the number of parts in any such partition of H must be given by a Wowzer-type function.
    Print ISSN: 0024-6115
    Electronic ISSN: 1460-244X
    Topics: Mathematics
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  • 2
    Publication Date: 2014-12-21
    Description: We have investigated earthquake source parameters and seismic moment-magnitude relations from 103 regional and local earthquakes with moment magnitude 2.6 to 7.2, which occurred in a distance range from 4.5 to 550 km during 1995–2012 by applying Brune’s seismic source model (J Geophys Res 75:4997–5009, 1970, J Geophys Res 76:5002, 1971) for P- and S/Lg-wave displacement spectra. Considering P- and S-wave data separately, we first studied the empirical dependence of the Fourier spectral amplitudes Ω due to the geometrical spreading and the inelastic attenuation and of the corner frequency, f0, with the epicentral distances, R. We found the distance correction parameters, Re0.0042R and R0.8333e0.00365R for the low-frequency spectral amplitudes and f0 = f0 ′e0.00043R and f0 = f0 ′e0.00044R for the corner frequency at the source, f0, and observed at the station, f0 ′, from P-wave and S-wave spectra, respectively. Applying the distance correction procedure, we determined the source displacement spectrum of P and S waves for each earthquake to estimate the seismic moment, M0; the moment magnitude, MW; the source radius, r; and the stress drop, Δσ. The seismic moments range from 1.06 × 1013 to 7.67 × 1019 N m, and their corresponding moment magnitudes are in the range of 2.6–7.2. Values of stress drop Δσ vary from 0.1 to 44 MPa. It was found that the stress drop increases with the increasing seismic moment in the range of 1013–1016 N m and possibly becomes constant at higher magnitudes, reaching a maximum value of about 40–45 MPa. We demonstrate that the values of the M0 and MW estimated from P-wave and S-wave analysis are consistent and confirmed by the results of waveform inversions, i.e., centroid moment tensor (CMT) solution. © 2014, Springer Science+Business Media Dordrecht.
    Print ISSN: 1383-4649
    Electronic ISSN: 1573-157X
    Topics: Geosciences , Physics
    Published by Springer
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