Distortion of boundary sets under inner functions. II

33 pages, no figures.-- MSC2000 codes: 32A30, 30C85, 30D50.MR#: MR1379286 (97b:30035)Zbl#: Zbl 0847.32005We present a study of the metric transformation properties of inner functions of several complex variables. Along the way we obtain fractional dimensional ergodic properties of classical inner functions.Publicad

Totally umbilical disks and applications to surfaces in three-dimensional homogeneous spaces

Following [Ch] and [dCF], we give sufficient conditions for a disk type surface, with piecewise smooth boundary, to be totally umbilical for a given Coddazi pair. As a consequence, we obtain rigidity results for surfaces in space forms and in homogeneous product spaces that generalizes some known results.Ministerio de Educación y Ciencia MTM2007-65249Ministerio de Educación y Ciencia MTM2007-64504Junta de Andalucía P06-FQM-01642Junta de Andalucía FQM32

Quantum Dynamics on the Worldvolume from Classical su(n) Cohomology

A key symmetry of classical $p$-branes is invariance under worldvolume diffeomorphisms. Under the assumption that the worldvolume, at fixed values of the time, is a compact, quantisable K\"ahler manifold, we prove that the Lie algebra of volume-preserving diffeomorphisms of the worldvolume can be approximated by $su(n)$, for $n\to\infty$. We also prove, under the same assumptions regarding the worldvolume at fixed time, that classical Nambu brackets on the worldvolume are quantised by the multibrackets corresponding to cocycles in the cohomology of the Lie algebra $su(n)$.Comment: This is a contribution to the Special Issue on Deformation Quantization, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Quantitative mixing results and inner functions

19 pages, no figures.-- MSC2000 codes: 30D05, 30D50, 37A05, 37A25, 37F10, 28D05, 11K55.MR#: MR2262783 (2007j:37003)Zbl#: Zbl 1125.30019We study in this paper estimates on the size of the sets of points which are well approximated by orbits of other points under certain dynamical systems. We apply the results obtained to the particular case of the dynamical system generated by inner functions in the unit disk of the complex plane.D. Pestana was supported by Grants BFM2003-04780 and BFM-2003-06335-C03-02, Ministerio de Ciencia y Tecnología, Spain. J. L. Fernández and M. V. Melián were supported by Grant BFM2003-04780 from Ministerio de Ciencia y Tecnología, Spain.Publicad

Hydrazones as Singular Reagents in Asymmetric Organocatalysis

This Minireview summarizes strategies and developments regarding the use of hydrazones as reagents in asymmetric organocatalysis, their distinct roles in nucleophile–electrophile, cycloaddition, and cyclization reactions. The key structural elements governing the reactivity of these reagents in a preferred pathway will be discussed, as well as their different interactions with organocatalysts, leading to diverse activation modes. Along these studies, the synthetic equivalence of N-monoalkyl, N,N-dialkyl, and N-acyl hydrazones with several synthons is also highlighted. Emphasis is also put on the mechanistic studies performed to understand the observed reactivities. Finally, the functional group transformations performed from the available products has also been analyzed, highlighting the synthetic value of these methodologies, which served to access numerous families of valuable multifunctional compounds and nitrogen-containing heterocycles.Ministerio de Economía y Competitividad CTQ2013-48164-C2-1-P, CTQ201348164-C2-2-PEuropean FEDER fundsJunta de Andalucía 2012/FQM 107

Efficient numerical schemes for viscoplastic avalanches. Part 2: the 2D case

This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. Here we derive such schemes in 2D as the follow up of the companion paper treating the 1D case. Numerical tests include in particular a generalized 2D benchmark for Bingham codes (the Bingham–Couette flow with two non-zero boundary conditions on the velocity) and a simulation of the avalanche path of Taconnaz in Chamonix—Mont-Blanc to show the usability of these schemes on real topographies from digital elevation models (DEM)