About a possible 3rd order phase transition at T=0 in 4D gluodynamics

We revisit the question of the convergence of lattice perturbation theory for a pure SU(3) lattice gauge theory in 4 dimensions. Using a series for the average plaquette up to order 10 in the weak coupling parameter beta^{-1}, we show that the analysis of the extrapolated ratio and the extrapolated slope suggests the possibility of a non-analytical power behavior of the form (1/\beta -1/5.7(1))^{1.0(1)}, in agreement with another analysis based on the same asumption. This would imply that the third derivative of the free energy density diverges near beta =5.7. We show that the peak in the third derivative of the free energy present on 4^4 lattices disappears if the size of the lattice is increased isotropically up to a 10^4 lattice. On the other hand, on 4 x L^3 lattices, a jump in the third derivative persists when L increases. Its location coincides with the onset of a non-zero average for the Polyakov loop. We show that the apparent contradiction at zero temperature can be resolved by moving the singularity in the complex 1/\beta plane. If the imaginary part of the location of the singularity Gamma is within the range 0.001< Gamma < 0.01, it is possible to limit the second derivative of P within an acceptable range without affecting drastically the behavior of the perturbative coefficients. We discuss the possibility of checking the existence of these complex singularities by using the strong coupling expansion or calculating the zeroes of the partition function.Comment: 7 pages, 9 figures, contains a resolution of the main paradox and a discussion of possible check

Phylogenies of atpD and recA support the small subunit rRNA-based classification of rhizobia

The current classification of the rhizobia (root-nodule symbionts) assigns them to six genera. It is strongly influenced by the small subunit (16S, SSU) rRNA molecular phylogeny, but such single-gene phylogenies may not reflect the evolution of the genome as a whole. To test this, parts of the atpD and recA genes have been sequenced for 25 type strains within the alpha -Proteobacteria, representing species in Rhizobium, Sinorhizobium, Mesorhizobium, Bradyrhizobium, Azorhizobium, Agrobacterium, Phyllobacterium, Mycoplana and Brevundimonas. The current genera Sinorhizobium and Mesorhizobium are well supported by these genes, each forming a distinct phylogenetic clade with unequivocal bootstrap support. There is good support for a Rhizobium clade that includes Agrobacterium tumefaciens, and the very close relationship between Agrobacterium rhizogenes and Rhizobium tropici is confirmed. There is evidence for recombination within the genera Mesorhizobium and Sinorhizobium, but the congruence of the phylogenies at higher levels indicates that the genera are genetically isolated. rRNA provides a reliable distinction between genera, but genetic relationships within a genus may be disturbed by recombination

A Stein characterisation of the distribution of the product of correlated normal random variables

We obtain a Stein characterisation of the distribution of the product of two correlated normal random variables with non-zero means, and more generally the distribution of the sum of independent copies of such random variables. Our Stein characterisation is shown to naturally generalise a number of other Stein characterisations in the literature. From our Stein characterisation we derive recursive formulas for the moments of the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, which allows for efficient computation of higher order moments.Comment: 12 page