Bivariate luminosity functions (LFs) are computed for galaxies in the New York Value-Added Galaxy Catalogue, based on the Sloan Digital Sky Survey Data Release 4. The galaxy properties investigated are the morphological type, inverse concentration index, Sersic index, absolute effective surface brightness, reference frame colours, absolute radius, eClass spectral type, stellar mass and galaxy environment. The morphological sample is flux-limited to galaxies with r < 15.9 and consists of 37,047 classifications to an RMS accuracy of +/- half a class in the sequence E, S0, Sa, Sb, Sc, Sd, Im. These were assigned by an artificial neural network, based on a training set of 645 eyeball classifications. The other samples use r < 17.77 with a median redshift of z ~ 0.08, and a limiting redshift of z < 0.15 to minimize the effects of evolution. Other cuts, for example in axis ratio, are made to minimize biases. A wealth of detail is seen, with clear variations between the LFs according to absolute magnitude and the second parameter. They are consistent with an early type, bright, concentrated, red population and a late type, faint, less concentrated, blue, star forming population. This bimodality suggests two major underlying physical processes, which in agreement with previous authors we hypothesize to be merger and accretion, associated with the properties of bulges and discs respectively. The bivariate luminosity-surface brightness distribution is fit with the Choloniewski function (a Schechter function in absolute magnitude and Gaussian in surface brightness). The fit is found to be poor, as might be expected if there are two underlying processes.
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