Higher-Order Properties of Analytic Wavelets

The influence of higher-order wavelet properties on the analytic wavelet transform behavior is investigated, and wavelet functions offering advantageous performance are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a two-parameter family of exactly analytic continuous wavelets. The degree of time/frequency localization, the existence of a mapping between scale and frequency, and the bias involved in estimating properties of modulated oscillatory signals, are proposed as important considerations. Wavelet behavior is found to be strongly impacted by the degree of asymmetry of the wavelet in both the frequency and the time domain, as quantified by the third central moments. A particular subset of the generalized Morse wavelets, recognized as deriving from an inhomogeneous Airy function, emerge as having particularly desirable properties. These "Airy wavelets" substantially outperform the only approximately analytic Morlet wavelets for high time localization. Special cases of the generalized Morse wavelets are examined, revealing a broad range of behaviors which can be matched to the characteristics of a signal.Comment: 15 pages, 6 Postscript figure

On the relation between sSFR and metallicity

In this paper we present an exact general analytic expression $Z(sSFR)=y/\Lambda(sSFR)+I(sSFR)$ linking the gas metallicity Z to the specific star formation rate (sSFR), that validates and extends the approximate relation put forward by Lilly et al. (2013, L13), where $y$ is the yield per stellar generation, $\Lambda(sSFR)$ is the instantaneous ratio between inflow and star formation rate expressed as a function of the sSFR, and $I$ is the integral of the past enrichment history, respectively. We then demonstrate that the instantaneous metallicity of a self-regulating system, such that its sSFR decreases with decreasing redshift, can be well approximated by the first term on the right-hand side in the above formula, which provides an upper bound to the metallicity. The metallicity is well approximated also by the L13 ideal regulator case, which provides a lower bound to the actual metallicity. We compare these approximate analytic formulae to numerical results and infer a discrepancy <0.1 dex in a range of metallicities and almost three orders of magnitude in the sSFR. We explore the consequences of the L13 model on the mass-weighted metallicity in the stellar component of the galaxies. We find that the stellar average metallicity lags 0.1-0.2 dex behind the gas-phase metallicity relation, in agreement with the data. (abridged)Comment: 14 pages, 6 figures, MNRAS accepte