Statistical properties of the Burgers equation with Brownian initial velocity

We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin x=0). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its $n$-point correlations. In the same limit, we derive the $n-$point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.Comment: 42 pages, published in J. Stat. Phy

Summing up the perturbation series in the Schwinger Model

Perturbation series for the electron propagator in the Schwinger Model is summed up in a direct way by adding contributions coming from individual Feynman diagrams. The calculation shows the complete agreement between nonperturbative and perturbative approaches.Comment: 10 pages (in REVTEX

Carotenoid content and reflectance of yellow and red nuptial plumages in widowbirds (Euplectes spp.)

1. Ornamental carotenoid coloration is commonly based on several different pigments with different nutritional and metabolic constraints. The identification and quantification of carotenoid pigments is therefore crucial to the understanding of signal content and signal evolution. 2. In male widowbirds (Euplectes spp.), the striking yellow and red carotenoid colours have been measured by reflectance spectrometry and studied with respect to sexual selection through male contest competition, but their biochemical mechanisms have not been analysed. 3. Here we use reflectance analysis and high performance liquid chromatography (HPLC) to describe the species-specific colours and plumage carotenoids in three widowbird species: yellow-mantled widowbird (YMW) Euplectes macrourus, red-shouldered widowbird (RSW) E. axillaris and red-collared widowbird (RCW) E. ardens. 4. YMW yellow (‘hue’ colorimetric λR50 = 522 nm) derives from the two ‘dietary yellow’ xanthophylls lutein and zeaxanthin, together with small amounts of ‘derived yellow’ pigments (3′-dehydrolutein and canary xanthophylls). 5. RCW red (λR50 = 574 nm) is achieved by the addition of low concentrations of ‘derived red ’ 4-keto-carotenoids, notably α- and β-doradexanthin and canthaxanthin. 6. RSW red (λR50 = 589 nm) is, in contrast, created by high concentrations of ‘dietary yellow ’ pigments (lutein, zeaxanthin) and ‘derived yellow ’ anhydrolutein, the latter only recently described in birds. 7. The two different mechanisms of producing red plumage are compared with other bird species and discussed with regard to costs and signal ‘honesty’