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Cerrado is the second largest biome in South America and accounted for the second largest contribution to carbon emissions in Brazil for the last 10 years, mainly due to land-use changes. It comprises approximately 2 million km2 and is divided into 22 ecoregions, based on environmental conditions and vegetation. The most dominant vegetation type is cerrado sensu stricto (cerrado ss), a savanna woodland. Quantifying variation of biomass density of this vegetation is crucial for climate change mitigation policies. Integrating remote sensing data with adequate allometric equations and field-based data sets can provide large-scale estimates of biomass. We developed individual-tree aboveground biomass (AGB) allometric models to compare different regression techniques and explanatory variables. We applied the model with the strongest fit to a comprehensive ground-based data set (77 sites, 893 plots, and 95,484 trees) to describe AGB density variation of cerrado ss. We also investigated the influence of physiographic and climatological variables on AGB density; this analysis was restricted to 68 sites because eight sites could not be classified into a specific ecoregion, and one site had no soil texture data. In addition, we developed two models to estimate plot AGB density based on plot basal area. Our data show that for individual-tree AGB models a) log-log linear models provided better estimates than nonlinear power models; b) including species as a random effect improved model fit; c) diameter at 30 cm above ground was a reliable predictor for individual-tree AGB, and although height significantly improved model fit, species wood density did not. Mean tree AGB density in cerrado ss was 22.9 tons ha-1 (95% confidence interval = ± 2.2) and varied widely between ecoregions (8.8 to 42.2 tons ha-1), within ecoregions (e.g. 4.8 to 39.5 tons ha-1), and even within sites (24.3 to 69.9 tons ha-1). Biomass density tended to be higher in sites close to the Amazon. Ecoregion explained 42% of biomass variation between the 68 sites (P < 0.01) and shows strong potential as a parameter for classifying regional biomass variation in the Cerrado. [ABSTRACT FROM AUTHOR]

We show how to obtain a local representation for Laguerre minimal surfaces based on vector fields in the unit two sphere that are defined in terms of a harmonic map in the complex plane. We give a geometric characterization of the Laguerre minimal surfaces constructed in the special cases where the harmonic map is either holomorphic or anti-holomorphic. [ABSTRACT FROM AUTHOR]

The article focuses on a method to optically find out Gaussian curvature k of a surface which is the product of the principal curvatures, k1 and k2, of the given point. Also a link between surfaces with constant k and area preserving maps between a sphere and a planar region has been found out, in this way a relationship is established between the branch of optics and geometry using Gauss map. Mirrored surface of a sphere is related to Lambert's azimuthal projection.