The strong Novikov conjecture for low degree cohomology

We show that for each discrete group G, the rational assembly map K_*(BG) \otimes Q \to K_*(C*_{max} G) \otimes \Q is injective on classes dual to the subring generated by cohomology classes of degree at most 2 (identifying rational K-homology and homology via the Chern character). Our result implies homotopy invariance of higher signatures associated to these cohomology classes. This consequence was first established by Connes-Gromov-Moscovici and Mathai. Our approach is based on the construction of flat twisting bundles out of sequences of almost flat bundles as first described in our previous work. In contrast to the argument of Mathai, our approach is independent of (and indeed gives a new proof of) the result of Hilsum-Skandalis on the homotopy invariance of the index of the signature operator twisted with bundles of small curvature.Comment: 11 page

Internal Energy of the Potts model on the Triangular Lattice with Two- and Three-body Interactions

We calculate the internal energy of the Potts model on the triangular lattice with two- and three-body interactions at the transition point satisfying certain conditions for coupling constants. The method is a duality transformation. Therefore we have to make assumptions on uniqueness of the transition point and that the transition is of second order. These assumptions have been verified to hold by numerical simulations for q=2, 3 and 4, and our results for the internal energy are expected to be exact in these cases.Comment: 9 pages, 4 figure

Monte Carlo Study of an Extended 3-State Potts Model on the Triangular Lattice

By introducing a chiral term into the Hamiltonian of the 3-state Potts model on a triangular lattice additional symmetries are achieved between the clockwise and anticlockwise states and the ferromagnetic state. This model is investigated using Monte Carlo methods. We investigate the full phase diagram and find evidence for a line tricritical points separating the ferromagnetic and antiferromagnetic phases.Comment: 6 pages, 10 figure

Localisation and colocalisation of KK-theory at sets of primes

Given a set of prime numbers S, we localise equivariant bivariant Kasparov theory at S and compare this localisation with Kasparov theory by an exact sequence. More precisely, we define the localisation at S to be KK^G(A,B) tensored with the ring of S-integers Z[S^-1]. We study the properties of the resulting variants of Kasparov theory.Comment: 16 page

Proof of Bose-Einstein Condensation for Dilute Trapped Gases

The ground state of bosonic atoms in a trap has been shown experimentally to display Bose-Einstein condensation (BEC). We prove this fact theoretically for bosons with two-body repulsive interaction potentials in the dilute limit, starting from the basic Schroedinger equation; the condensation is 100% into the state that minimizes the Gross-Pitaevskii energy functional. This is the first rigorous proof of BEC in a physically realistic, continuum model.Comment: Revised version with some simplifications and clarifications. To appear in Phys. Rev. Let

Normalized indices derived from visceral adipose mass assessed by MRI and their correlation with markers for insulin resistance and prediabetes

Visceral adipose tissue (VAT) plays an important role in the pathogenesis of insulin resistance (IR), prediabetes and type 2 diabetes. However, VAT volume alone might not be the best marker for insulin resistance and prediabetes or diabetes, as a given VAT volume may differently impact on these metabolic traits based on body height, gender, age and ethnicity. In a cohort of 1295 subjects from the Tübingen Diabetes Family Study (TDFS) and in 9978 subjects from the UK Biobank (UKBB), undergoing magnetic resonance imaging for quantification of VAT volume, total adipose tissue (TAT, in the TDFS), total abdominal adipose tissue (TAAT) in the UKBB, and total lean tissue (TLT), VAT volume and several VAT-indices were investigated for their relationships with insulin resistance and glycemic traits. VAT-related indices were calculated by correcting for body height (VAT/m: VAT/body height; VAT/m²: VAT/(body height)², and VAT/m³: VAT/(body height)³), TAT (%VAT), TLT (VAT/TLT) and weight (VAT/WEI), with closest equivalents used within the UKBB dataset. Prognostic values of VAT and VAT-related indices for insulin sensitivity, HbA1c levels and prediabetes/diabetes were analyzed for males and females. Males had higher VAT volume and VAT-related indices than females in both cohorts (p < 0.0001) and VAT volume has shown to be a stronger determinant for insulin sensitivity than anthropometric variables. Among the parameters uncorrected VAT and derived indices, VAT/m³ most strongly correlated negatively with insulin sensitivity and positively with HbA1c levels and prediabetes/diabetes in the TDFS (R² = 0.375/0.305 for females/males for insulin sensitivity, 0.178/0.148 for HbA1c levels vs. – e.g. – 0.355/0.293 and 0.144/0.133 for VAT, respectively) and positively with HbA1c (R² = 0.046/0.042) in the UKBB for females and males. Furthermore, VAT/m³ was found to be a significantly better determinant of insulin resistance or prediabetes than uncorrected VAT volume (p < 0.001/0.019 for females/males regarding insulin sensitivity, p < 0.001/< 0.001 for females/males regarding HbA1c). Evaluation of several indices derived from VAT volume identified VAT/m³ to most strongly correlate with insulin sensitivity and glucose metabolism. Thus, VAT/m³ appears to provide better indications of metabolic characteristics (insulin sensitivity and pre-diabetes/diabetes) than VAT volume alone

One-dimensional Kondo lattice at partial band filling

An effective Hamiltonian for the localized spins in the one-dimensional Kondo lattice model is derived via a unitary transformation involving a bosonization of delocalized conduction electrons. The effective Hamiltonian is shown to reproduce all the features of the model as identified in various numerical simulations, and provides much new information on the ferro- to paramagnetic phase transition and the paramagnetic phase.Comment: 11 pages Revtex, 1 Postscript figure. To appear in Phys. Rev. Let

Hodge Theory on Metric Spaces

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. The appendix by Anthony Baker provides a separable, compact metric space with infinite dimensional \alpha-scale homology.Comment: appendix by Anthony W. Baker, 48 pages, AMS-LaTeX. v2: final version, to appear in Foundations of Computational Mathematics. Minor changes and addition

Free expansion of two-dimensional condensates with a vortex

We study the free expansion of a pancake-shaped Bose-condensed gas, which is initially trapped under harmonic confinement and containing a vortex at its centre. In the case of a radial expansion holding fixed the axial confinement we consider various models for the interactions, depending on the thickness of the condensate relative to the value of the scattering length. We are thus able to evaluate different scattering regimes ranging from quasi-three-dimensional (Q3D) to strictly two-dimensional (2D). We find that as the system goes from Q3D to 2D the expansion rate of the condensate increases whereas that of the vortex core decreases. In the Q3D scattering regime we also examine a fully free expansion in 3D and find oscillatory behaviour for the vortex core radius: an initial fast expansion of the vortex core is followed by a slowing down. Such a nonuniform expansion rate of the vortex core may be taken into account in designing new experiments.Comment: 10 pages, 4 figure