Census of Planar Maps: From the One-Matrix Model Solution to a Combinatorial Proof

We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an alternative and purely combinatorial solution to the problem of counting arbitrary planar maps with prescribed vertex degrees.Comment: 29 pages, 14 figures, tex, harvmac, eps

All-Helicity Symbol Alphabets from Unwound Amplituhedra

We review an algorithm for determining the branch points of general amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills theory from amplituhedra. We demonstrate how to use the recent reformulation of amplituhedra in terms of `sign flips' in order to streamline the application of this algorithm to amplitudes of any helicity. In this way we recover the known branch points of all one-loop amplitudes, and we find an `emergent positivity' on boundaries of amplituhedra.Comment: 38 pages, 5 figures, 1 big table; v2: minor corrections and improvement

Classification and analysis of emission-line galaxies using mean field independent component analysis

We present an analysis of the optical spectra of narrow emission-line galaxies, based on mean field independent component analysis (MFICA). Samples of galaxies were drawn from the Sloan Digital Sky Survey (SDSS) and used to generate compact sets of `continuum' and `emission-line' component spectra. These components can be linearly combined to reconstruct the observed spectra of a wider sample of galaxies. Only 10 components - five continuum and five emission line - are required to produce accurate reconstructions of essentially all narrow emission-line galaxies; the median absolute deviations of the reconstructed emission-line fluxes, given the signal-to-noise ratio (S/N) of the observed spectra, are 1.2-1.8 sigma for the strong lines. After applying the MFICA components to a large sample of SDSS galaxies we identify the regions of parameter space that correspond to pure star formation and pure active galactic nucleus (AGN) emission-line spectra, and produce high S/N reconstructions of these spectra. The physical properties of the pure star formation and pure AGN spectra are investigated by means of a series of photoionization models, exploiting the faint emission lines that can be measured in the reconstructions. We are able to recreate the emission line strengths of the most extreme AGN case by assuming the central engine illuminates a large number of individual clouds with radial distance and density distributions, f(r) ~ r^gamma and g(n) ~ n^beta, respectively. The best fit is obtained with gamma = -0.75 and beta = -1.4. From the reconstructed star formation spectra we are able to estimate the starburst ages. These preliminary investigations serve to demonstrate the success of the MFICA-based technique in identifying distinct emission sources, and its potential as a tool for the detailed analysis of the physical properties of galaxies in large-scale surveys.Comment: MNRAS accepted. 29 pages, 24 figures, 3 table