Large sets of objects with spectroscopic redshift measurements will be needed for imaging dark energy experiments to achieve their full potential, serving two goals:_training_, i.e., the use of objects with known redshift to develop and optimize phot
ometric redshift algorithms; and_calibration_, i.e., the characterization of moments of redshift (or photo-z error) distributions. Better training makes cosmological constraints from a given experiment stronger, while highly-accurate calibration is needed for photo-z systematics not to dominate errors. In this white paper, we investigate the required scope of spectroscopic datasets which can serve both these purposes for ongoing and next-generation dark energy experiments, as well as the time required to obtain such data with instruments available in the next decade. Large time allocations on kilo-object spectrographs will be necessary, ideally augmented by infrared spectroscopy from space. Alternatively, precision calibrations could be obtained by measuring cross-correlation statistics using samples of bright objects from a large baryon acoustic oscillation experiment such as DESI. We also summarize the additional work on photometric redshift methods needed to prepare for ongoing and future dark energy experiments.
Genetic feedback loops in cells break detailed balance and involve bimolecular reactions; hence exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. We here consider the master equation for a gene regulatory
feedback loop: a gene produces protein which then binds to the promoter of the same gene and regulates its expression. The protein degrades in its free and bound forms. This network breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. The full parametric dependence of the analytical non-equilibrium steady-state probability distribution is verified by direct numerical solution of the master equations. For the case where the degradation rate of bound and free protein is the same, our solution is at variance with a previous claim of an exact solution (Hornos et al, Phys. Rev. E {bf 72}, 051907 (2005) and subsequent studies). We show explicitly that this is due to an unphysical formulation of the underlying master equation in those studies.
