Einstein-like geometric structures on surfaces

An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenerate immersed hypersurface in flat affine space. The author has defined for AH structures Einstein equations, which specialize on the one hand to the usual Einstein Weyl equations and, on the other hand, to the equations for affine hyperspheres. Here these equations are solved for Riemannian signature AH structures on compact orientable surfaces, the deformation spaces of solutions are described, and some aspects of the geometry of these structures are related. Every such structure is either Einstein Weyl (in the sense defined for surfaces by Calderbank) or is determined by a pair comprising a conformal structure and a cubic holomorphic differential, and so by a convex flat real projective structure. In the latter case it can be identified with a solution of the Abelian vortex equations on an appropriate power of the canonical bundle. On the cone over a surface of genus at least two carrying an Einstein AH structure there are Monge-Amp`ere metrics of Lorentzian and Riemannian signature and a Riemannian Einstein K"ahler affine metric. A mean curvature zero spacelike immersed Lagrangian submanifold of a para-K"ahler four-manifold with constant para-holomorphic sectional curvature inherits an Einstein AH structure, and this is used to deduce some restrictions on such immersions.

Una casa para Einstein: Konrad Wachsmann y la evolución de un modelo prefabricado desde las casas "Christoph & Unmack A. G." al "General Panel System"

Este artículo estudia la evolución del modelo de vivienda prefabricada en madera que construye Konrad Wachsmann para Einstein en 1929 en Caputh, cerca de Potsdam. El físico deseaba construirse un "lugar de descanso", eligiendo la construcción en madera por su facilidad y rapidez de montaje, adaptabilidad, calidez y para que armonizara mejor con el medio ambiente. Wachsmann, que trabajaba para la firma "Christoph & Unmack A.G." le presentará un modelo prefabricado moderno. Esta tipología, evolucionada desde los diseños "nórdico-escandinavo" y "jugendstil", hasta introducir un nuevo lenguaje de líneas puras, cubierta plana, y grandes ventanales, será ligeramente modificada por Einstein, que finalmente adjudica el encargo. Wachsmann continuará la labor de investigación sobre vivienda prefabricada junto con Gropius en EEUU, que dará como resultado el "General Panel System" y sus conocidas "Packaged Houses".

Ricci flows on surfaces related to the Einstein weyl and Abelian vortex equations

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve these equations is preserved by the Ricci flow. The equations are solved explicitly, and among the metrics obtained are all steady gradient Ricci solitons (e.g. the cigar soliton) and the sausage metric; there are found other examples of eternal, ancient, and immortal Ricci flows, as well as some Ricci flows with conical singularities.

First-order equivalent to Einstein-Hilbert Lagrangian

A first-order Lagrangian L ∇ variationally equivalent to the second-order Einstein- Hilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by L ∇ is proved to be regular and its Hamiltonian formulation is studied, including its covariant Hamiltonian attached to ∇ .

Variation in per capita interaction strength: Thresholds due to nonlinear dynamics and nonequilibrium conditions

I measured the strength of interaction between a marine herbivore and its growing resource over a realistic range of absolute and relative abundances. The herbivores (hermit crabs: Pagurus spp.) have slow and/or weak functional and numerical responses to epiphytic diatoms (Isthmia nervosa), which show logistic growth in the absence of consumers. By isolating this interaction in containers in the field, I mimicked many of the physical and biological variables characteristic of the intertidal while controlling the densities of focal species. The per capita effects of consumers on the population dynamics of their resource (i.e., interaction strength) were defined by using the relationship between hermit crab density and proportional change in the resource. When this relationship is fit by a Weibull function, a single parameter distinguishes constant interaction strength from one that varies as a function of density. Constant interaction strength causes the proportion of diatoms to fall linearly or proportionally as hermit crab density increases whereas per capita effects that increase with density cause an accelerating decline. Although many mathematical models of species interactions assume linear dynamics and invariant parameters, at least near equilibrium, the per capita effects of hermit crabs on diatoms varied substantially, apparently crossing a threshold from weak to strong when consumption exceeded resource production. This threshold separates a domain of coexistence from one of local extinction of the resource. Such thresholds may help explain trophic cascades, resource compensation, and context-dependent interaction strengths, while indicating a way to predict trophic effects, despite nonlinearities, as a function of vital rates.

Nonequilibrium mechanism of transcription termination from observations of single RNA polymerase molecules

Cessation of transcription at specific terminator DNA sequences is used by viruses, bacteria, and eukaryotes to regulate the expression of downstream genes, but the mechanisms of transcription termination are poorly characterized. To elucidate the kinetic mechanism of termination at the intrinsic terminators of enteric bacteria, we observed, by using single-molecule light microscopy techniques, the behavior of surface-immobilized Escherichia coli RNA polymerase (RNAP) molecules in vitro. An RNAP molecule remains at a canonical intrinsic terminator for ≈64 s before releasing DNA, implying the formation of an elongation-incompetent (paused) intermediate by transcription complexes that terminate but not by those that read through the terminator. Analysis of pause lifetimes establishes a complete minimal mechanism of termination in which paused intermediate formation is both necessary and sufficient to induce release of RNAP at the terminator. The data suggest that intrinsic terminators function by a nonequilibrium process in which terminator effectiveness is determined by the relative rates of nucleotide addition and paused state entry by the transcription complex.

Free energy reconstruction from nonequilibrium single-molecule pulling experiments

Laser tweezers and atomic force microscopes are increasingly used to probe the interactions and mechanical properties of individual molecules. Unfortunately, using such time-dependent perturbations to force rare molecular events also drives the system away from equilibrium. Nevertheless, we show how equilibrium free energy profiles can be extracted rigorously from repeated nonequilibrium force measurements on the basis of an extension of Jarzynski's remarkable identity between free energies and the irreversible work.

Groups, information theory, and Einstein's likelihood principle

We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to the composition of independent systems, in the context of a recently proposed generalization of the Shannon-Khinchin axioms. We associate to each member of a large class of entropies a generalized information measure, satisfying the additivity property on a set of independent systems as a consequence of the underlying group law. At the same time, we also show that Einstein's likelihood function naturally emerges as a byproduct of our informational interpretation of (generally nonadditive) entropies. These results confirm the adequacy of composable entropies both in physical and social science contexts.

Nonequilibrium spin glass dynamics with Janus

The out of equilibrium evolution for an Edwards‐Anderson spin glass is followed for a tenth of a second, a long enough time to let us make safe predictions about the behaviour at experimental scales. This work has been made possible by Janus, an FPGA based special purpose computer. We have thoroughly studied the spin glass correlation functions and the growth of the coherence length for L = 80 lattices in 3D. Our main conclusion is that these spin glasses follow noncoarsening dynamics, at least up to the experimentally relevant time scales.

Nonequilibrium spin-glass dynamics from picoseconds to a tenth of a second

We study numerically the nonequilibrium dynamics of the Ising spin glass, for a time spanning 11 orders of magnitude, thus approaching the experimentally relevant scale (i.e., seconds). We introduce novel analysis techniques to compute the coherence length in a model-independent way. We present strong evidence for a replicon correlator and for overlap equivalence. The emerging picture is compatible with noncoarsening behavior.