Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle.     

Towards a Microscopic Understanding of Nucleon Polarizabilities

We outline a microscopic framework to calculate nucleon Compton scattering from the level of quarks and gluons within the covariant Faddeev approach. We explain the connection with hadronic expansions of the Compton scattering amplitude and discuss the obstacles in maintaining electromagnetic gauge invariance. Finally we give preliminary results for the nucleon polarizabilities.     

Unquenched complex dirac spectra at nonzero chemical potential: two-color QCD lattice data versus matrix model

We compare analytic predictions of non-Hermitian chiral random matrix theory with the complex Dirac operator eigenvalue spectrum of two-color lattice gauge theory with dynamical fermions at nonzero chemical potential. The Dirac eigenvalues come in complex conjugate pairs, making the action of this theory real and positive for our choice of two staggered flavors. This enables us to use standard Monte Carlo simulations in testing the influence of the chemical potential and quark mass on complex eigenvalues close to the origin. We find excellent agreement between the analytic predictions and our data for two different volumes over a range of chemical potentials below the chiral phase transition. In particular, we detect the effect of unquenching when going to very small quark masses.     

Non-existence of stationary two-black-hole configurations

We resume former discussions of the question, whether the spin–spin repulsion and the gravitational attraction of two aligned black holes can balance each other. To answer the question we formulate a boundary value problem for two separate (Killing-) horizons and apply the inverse (scattering) method to solve it. Making use of results of Manko, Ruiz and Sanabria-Gómez and a novel black hole criterion, we prove the non-existence of the equilibrium situation in question.     

The nature of the chemical bond revisited. An energy partitioning analysis of diatomic molecules E2 (E=N–Bi, F–I), CO and BF

The nature of the chemical bonds in the diatomic molecules E₂ (E=N–Bi, F–I), CO and BF has been studied with an energy partitioning analysis using gradient-corrected density functional theory calculations. The results make it possible to estimate quantitatively the strength of covalent and electrostatic attractions and the Pauli repulsion between the atoms. The data suggest that some traditional explanations regarding the strength of the molecules should be modified. The energy partitioning analysis shows that the chemical bonds in the group 15 diatomic molecules have significant electrostatic character, which increases from 30.1% in N₂ to 58.3% in Bi₂. The contribution of the electrostatic attraction to the binding interactions in Sb₂ and Bi₂ is larger than the covalent bonding. The strength of the bonding in the triply bonded dinitrogen is less than that of the bonding. The calculations indicate that E is between 32.2% (Bi₂) and 40.0% (P₂) of the total orbital interaction energy (E_orb). The much stronger bond of N₂, as compared with the heavier group 15 E₂ homologues, is not caused by a particularly strong contribution by the bonding, but rather by the relatively large interactions. The comparison of N₂ with isoelectronic CO shows that the electrostatic character in the heteroatomic molecule is slightly smaller (28.8%) than in the homoatomic molecule. The contribution of the bonding in CO is larger (49.2%) than in N₂ (34.3%). The reason why CO has a stronger bond than N₂ is the significantly weaker Pauli repulsion in CO. The electrostatic character of the bonding in BF is slightly larger (32.0%) than in CO and N₂. BF has much weaker -bonding contributions that provide only 11.2% of the covalent interactions, which is why BF has a much weaker bond than CO and N₂. The chemical bonds in the dihalogen molecules have much higher covalent than electrostatic character. The E_orb term contributes between 74.4% (Br₂) and 79.7% (F₂) to the total attractive interactions. The relatively weak bond in F₂ comes from the rather large Pauli repulsion.Contribution to the Jacopo Tomasi Honorary Issue