We discuss methods based on Principal Component Analysis to constrain the dark energy equation of state using a combination of Type Ia supernovae at low redshift and spectroscopic measurements of varying fundamental couplings at higher redshifts. We discuss the performance of this method when future better-quality datasets are available, focusing on two forthcoming ESO spectrographs -ESPRESSO for the VLT and CODEX for the E-ELT -which include these measurements as a key part of their science cases. These can realize the prospect of a detailed characterization of dark energy properties almost all the way up to redshift 4.
We define and study a class of (random) Boolean constraint satisfaction problems representing minimal feasibility constraints for networks of chemical reactions. The constraints we consider encode, respectively, for hard mass-balance conditions (where the consumption and production fluxes of each chemical species are matched) and for soft mass-balance conditions (where a net production of compounds is in principle allowed). We solve these constraint satisfaction problems under the Bethe approximation and derive the corresponding Belief Propagation equations, that involve 8 different messages. The statistical properties of ensembles of random problems are studied via the population dynamics methods. By varying a chemical potential attached to the activity of reactions, we find first order transitions and strong hysteresis, suggesting a non-trivial structure in the space of feasible solutions.
Contents
We analyze the solutions, on single network instances, of a recently introduced class of constraint-satisfaction problems (CSPs), describing feasible steady states of chemical reaction networks. First, we show that the CSPs generalize the scheme known as network expansion, which is recovered in a specific limit. Next, a full statistical mechanics characterization (including the phase diagram and a discussion of the physical origin of the phase transitions) for network expansion is obtained. Finally, we provide a message-passing algorithm to solve the original CSPs in the most general form.
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