Higher su(N) tensor products

We extend our recent results on ordinary su(N) tensor product multiplicities to higher su(N) tensor products. Particular emphasis is put on four-point couplings where the tensor product of four highest weight modules is considered. The number of times the singlet occurs in the decomposition is the associated multiplicity. In this framework, ordinary tensor products correspond to three-point couplings. As in that case, the four-point multiplicity may be expressed explicitly as a multiple sum measuring the discretised volume of a convex polytope. This description extends to higher-point couplings as well. We also address the problem of determining when a higher-point coupling exists, i.e., when the associated multiplicity is non-vanishing. The solution is a set of inequalities in the Dynkin labels.Comment: 17 pages, LaTe

Note on SLE and logarithmic CFT

It is discussed how stochastic evolutions may be linked to logarithmic conformal field theory. This introduces an extension of the stochastic Loewner evolutions. Based on the existence of a logarithmic null vector in an indecomposable highest-weight module of the Virasoro algebra, the representation theory of the logarithmic conformal field theory is related to entities conserved in mean under the stochastic process.Comment: 10 pages, LaTeX, v2: version to be publishe

A Girsanov approach to slow parameterizing manifolds in the presence of noise

We consider a three-dimensional slow-fast system with quadratic nonlinearity and additive noise. The associated deterministic system of this stochastic differential equation (SDE) exhibits a periodic orbit and a slow manifold. The deterministic slow manifold can be viewed as an approximate parameterization of the fast variable of the SDE in terms of the slow variables. In other words the fast variable of the slow-fast system is approximately "slaved" to the slow variables via the slow manifold. We exploit this fact to obtain a two dimensional reduced model for the original stochastic system, which results in the Hopf-normal form with additive noise. Both, the original as well as the reduced system admit ergodic invariant measures describing their respective long-time behaviour. We will show that for a suitable metric on a subset of the space of all probability measures on phase space, the discrepancy between the marginals along the radial component of both invariant measures can be upper bounded by a constant and a quantity describing the quality of the parameterization. An important technical tool we use to arrive at this result is Girsanov's theorem, which allows us to modify the SDEs in question in a way that preserves transition probabilities. This approach is then also applied to reduced systems obtained through stochastic parameterizing manifolds, which can be viewed as generalized notions of deterministic slow manifolds.Comment: 54 pages, 6 figure

Workshop on Drought Forecasting for Northeast Brazil

Precipitation forecasting parameters for northeast Brazil were developed. Hydrological, sociological, and economic aspects were examined. A drought forecasting model is presented

Greenalite Nanoparticles in Alkaline Vent Plumes as Templates for the Origin of Life

Mineral templates are thought to have played keys roles in the emergence of life. Drawing on recent findings from 3.45–2.45 billion-year-old iron-rich hydrothermal sedimentary rocks, we hypothesize that greenalite (Fe₃Si₂O₅ (OH)₄) was a readily available mineral in hydrothermal environments, where it may have acted as a template and catalyst in polymerization, vesicle formation and encapsulation, and protocell replication. We argue that venting of dissolved Fe²⁺ and SiO₂ (aq) into the anoxic Hadean ocean favored the precipitation of nanometer-sized particles of greenalite in hydrothermal plumes, producing a continuous flow of free-floating clay templates that traversed the ocean. The mixing of acidic, metal-bearing hydrothermal plumes from volcanic ridge systems with more alkaline, organic-bearing plumes generated by serpentinization of ultramafic rocks brought together essential building blocks for life in solutions conducive to greenalite precipitation. We suggest that the extreme disorder in the greenalite crystal lattice, producing structural modulations resembling parallel corrugations (∼22 Å wide) on particle edges, promoted the assembly and alignment of linear RNA-type molecules (∼20 Å diameter). In alkaline solutions, greenalite nanoparticles could have accelerated the growth of membrane vesicles, while their encapsulation allowed RNA-type molecules to continue to form on the mineral templates, potentially enhancing the growth and division of primitive cell membranes. Once self-replicating RNA evolved, the mineral template became redundant, and protocells were free to replicate and roam the ocean realm

Greenalite Nanoparticles in Alkaline Vent Plumes as Templates for the Origin of Life

Mineral templates are thought to have played keys roles in the emergence of life. Drawing on recent findings from 3.45–2.45 billion-year-old iron-rich hydrothermal sedimentary rocks, we hypothesize that greenalite (Fe₃Si₂O₅ (OH)₄) was a readily available mineral in hydrothermal environments, where it may have acted as a template and catalyst in polymerization, vesicle formation and encapsulation, and protocell replication. We argue that venting of dissolved Fe²⁺ and SiO₂ (aq) into the anoxic Hadean ocean favored the precipitation of nanometer-sized particles of greenalite in hydrothermal plumes, producing a continuous flow of free-floating clay templates that traversed the ocean. The mixing of acidic, metal-bearing hydrothermal plumes from volcanic ridge systems with more alkaline, organic-bearing plumes generated by serpentinization of ultramafic rocks brought together essential building blocks for life in solutions conducive to greenalite precipitation. We suggest that the extreme disorder in the greenalite crystal lattice, producing structural modulations resembling parallel corrugations (∼22 Å wide) on particle edges, promoted the assembly and alignment of linear RNA-type molecules (∼20 Å diameter). In alkaline solutions, greenalite nanoparticles could have accelerated the growth of membrane vesicles, while their encapsulation allowed RNA-type molecules to continue to form on the mineral templates, potentially enhancing the growth and division of primitive cell membranes. Once self-replicating RNA evolved, the mineral template became redundant, and protocells were free to replicate and roam the ocean realm

Self-trapping of impurities in Bose-Einstein condensates: Strong attractive and repulsive coupling

We study the interaction-induced localization -- the so-called self-trapping -- of a neutral impurity atom immersed in a homogeneous Bose-Einstein condensate (BEC). Based on a Hartree description of the BEC we show that -- unlike repulsive impurities -- attractive impurities have a singular ground state in 3d and shrink to a point-like state in 2d as the coupling approaches a critical value. Moreover, we find that the density of the BEC increases markedly in the vicinity of attractive impurities in 1d and 2d, which strongly enhances inelastic collisions between atoms in the BEC. These collisions result in a loss of BEC atoms and possibly of the localized impurity itself.Comment: 7 pages, 5 figure