We discuss the effects of quenched disorder in a dilute Bose-Einstein condensate confined in a hard walls trap. Starting from the disordered Gross-Pitaevskii functional, we obtain a representation for the quenched free energy as a series of integer moments of the partition function. Positive and negative disorder-dependent effective coupling constants appear in the integer moments. Going beyond the mean-field approximation, we compute the static two-point correlation functions at first-order in the positive effective coupling constants. We obtain the combined contributions of effects due to boundary conditions and disorder in this weakly disordered condensate. The ground state renormalized density profile of the condensate is presented. We also discuss the appearance of metastable and true ground states for strong disorder, when the effective coupling constants become negative.

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We discuss modifications in the integral representation of the Riemann zetafunction that lead to generalizations of the Riemann functional equation that preserves the symmetry s → (1 − s) in the critical strip. By modifying one integral representation of the zeta-function with a cut-off that does exhibit the symmetry x → 1/x, we obtain a generalized functional equation involving Bessel functions of second kind. Next, with another cut-off that does exhibit the same symmetry, we obtain a generalization for the functional equation involving only one Bessel function of second kind. Some connection between one regularized zeta-function and the Laplace transform of the heat kernel for the Euclidean and hyperbolic space is discussed.

The performance of an outdoor pilot-scale system consisting of a high rate algal pond interconnected to an absorption bubble column and seeded with a microalgal-bacterial consortium is presented. Biogas and organic leachate (from food waste anaerobic hydrolysis) were used as sources of CO2 and nutrients for the consortium, respectively. The removal efficiencies for CO2 and H2S from biogas were 80.0% and 99.9%, respectively, provoking a methane content in the utilized biogas of approximately 55 vol% for anaerobically-produced biogas and 90 vol% for a prepared gaseous mixture. The molecular oxygen produced by microalgae photosynthesis induced undesirable cumulation of O2 in utilized biogas up to 5%vol. The organic leachate was diluted in an alkaline mineral medium (15:1 v/v), and this rich-nutrient mixture was fed to the system at a hydraulic retention time of 138 d. The biomass productivity applying organic leachate and anaerobically-produced biogas was 2.6 g m−3 d−1, nine times lower than biomass productivity achieved when only mineral salt medium was fed, suggesting the existence of limited nutrient bioavailability. Microalgal-bacterial community shifts evaluated through molecular and morphological methods showed that a robust and stable photosynthetic population was maintained throughout the entire operation of the system, formed mainly by Picochlorum sp., Pseudanabaena sp., Spirulina sp., and Nitzschia sp., as well as heterotrophic bacteria and some microalgae predators. The valued by-products after waste utilization were clean and highly calorific biogas besides microalgae biomass, where the commercialization of the latter would determine the economic viability of biogas and leachate utilization. Graphical Abstract

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