A comparative Digital Soil Mapping (DSM) study using a non-supervised clustering analysis and an expert knowledge based model - A case study from Ahuachapán, El Salvador

DSM is the inference of spatial and temporal soil property variations using mathematical models based on quantitative relationships between environmental information and soil measurements. The quality of DSM information depends on the method and environmental covariates used for its estimations. We compared two DSM methods to predict soil properties such as Organic Matter “MO” (%), Sand (%), Clay (%), pH (H2O), Phosphorus (mg/kg), Effective Cationic Exchange Capacity “CICE” (cmol/L), Potassium (cmol/L) and Water Holding Capacity (mm/m) for the department of Ahuachapán in El Salvador to support the activities of the Agriculture Landscape Restoration Initiative (ALRI) in the countr

Catenaries and minimal surfaces of revolution in hyperbolic space

We introduce the concept of extrinsic catenary in the hyperbolic plane. Working in the hyperboloid model, we define an extrinsic catenary as the shape of a curve hanging under its weight as seen from the ambient space. In other words, an extrinsic catenary is a critical point of the potential functional, where we calculate the potential with the extrinsic distance to a fixed reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and as a solution to a prescribed curvature problem involving certain vector fields. In addition, we prove that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary with respect to an appropriate reference plane. Finally, we prove that one of the families of extrinsic catenaries admits an intrinsic characterization if we replace the extrinsic distance with the intrinsic length of horocycles orthogonal to a reference geodesic

A comparative Digital Soil Mapping (DSM) study using a non-supervised clustering analysis and an expert knowledge based model - A case study from Ahuachapán, El Salvador

DSM is the inference of spatial and temporal soil property variations using mathematical models based on quantitative relationships between environmental information and soil measurements. The quality of DSM information depends on the method and environmental covariates used for its estimations. We compared two DSM methods to predict soil properties such as Organic Matter “MO” (%), Sand (%), Clay (%), pH (H2O), Phosphorus (mg/kg), Effective Cationic Exchange Capacity “CICE” (cmol/L), Potassium (cmol/L) and Water Holding Capacity (mm/m) for the department of Ahuachapán in El Salvador to support the activities of the Agriculture Landscape Restoration Initiative (ALRI) in the countr

Catenaries in Riemannian surfaces

The concept of catenary has been recently extended to the sphere and the hyperbolic plane by the second author (López, arXiv:2208.13694). In this work, we define catenaries on any Riemannian surface. A catenary on a surface is a critical point of the potential functional, where we calculate the potential with the intrinsic distance to a fixed reference geodesic. Adopting semi-geodesic coordinates around the reference geodesic, we characterize catenaries using their curvature. Finally, after revisiting the space-form catenaries, we consider surfaces of revolution (where a Clairaut relation is established), ruled surfaces, and the Grušin plane.</p

Catenaries and minimal surfaces of revolution in hyperbolic space

We introduce the concept of extrinsic catenary in the hyperbolic plane. Working in the hyperboloid model, we define an extrinsic catenary as the shape of a curve hanging under its weight as seen from the ambient space. In other words, an extrinsic catenary is a critical point of the potential functional, where we calculate the potential with the extrinsic distance to a fixed reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and as a solution to a prescribed curvature problem involving certain vector fields. In addition, we prove that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary with respect to an appropriate reference plane. Finally, we prove that one of the families of extrinsic catenaries admits an intrinsic characterization if we replace the extrinsic distance with the intrinsic length of horocycles orthogonal to a reference geodesic

Catenaries and minimal surfaces of revolution in hyperbolic space

We introduce the concept of extrinsic catenary in the hyperbolic plane. Working in the hyperboloid model, we define an extrinsic catenary as the shape of a curve hanging under its weight, where the weight is the distance with respect to a reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and also as a solution to a prescribed curvature problem involving certain vector fields. Finally, we prove that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary with respect to an appropriate reference plane.Comment: 17 pages. Keywords: Catenary, extrinsic catenary, hyperbolic space, minimal surface, surface of revolutio

Catenaries and singular minimal surfaces in the simply isotropic space

This paper investigates the hanging chain problem in the simply isotropic plane as well as its 2-dimensional analog in the simply isotropic space. The simply isotropic plane and space are two- and three-dimensional geometries equipped with a degenerate metric whose kernel has dimension 1. Although the metric is degenerate, the hanging chain and hanging surface problems are well-posed if we employ the relative arc length and relative area to measure the weight. Here, the concepts of relative arc length and relative area emerge by seeing the simply isotropic geometry as a relative geometry. In addition to characterizing the simply isotropic catenary, i.e., the solutions of the hanging chain problem, we also prove that it is the generating curve of a minimal surface of revolution in the simply isotropic space. Finally, we obtain the 2-dimensional analog of the catenary, the so-called singular minimal surfaces, and determine the shape of a hanging surface of revolution in the simply isotropic space.Comment: 21 pages, 2 figures; Keywords: Simply isotropic space, catenary, singular minimal surface, relative geometr

Catenaries in Riemannian surfaces

The concept of catenary has been recently extended to the sphere and the hyperbolic plane by the second author (López, arXiv:2208.13694). In this work, we define catenaries on any Riemannian surface. A catenary on a surface is a critical point of the potential functional, where we calculate the potential with the intrinsic distance to a fixed reference geodesic. Adopting semi-geodesic coordinates around the reference geodesic, we characterize catenaries using their curvature. Finally, after revisiting the space-form catenaries, we consider surfaces of revolution (where a Clairaut relation is established), ruled surfaces, and the Grušin plane.</p

Catenaries in Riemannian Surfaces

The concept of catenary has been recently extended to the sphere and the hyperbolic plane by the second author [López, arXiv:2208.13694]. In this work, we define catenaries on any Riemannian surface. A catenary on a surface is a critical point of the potential functional, where we calculate the potential with the intrinsic distance to a fixed reference geodesic. Adopting semi-geodesic coordinates around the reference geodesic, we characterize catenaries using their curvature. Finally, after revisiting the space-form catenaries, we consider surfaces of revolution (where a Clairaut relation is established), ruled surfaces, and the Grušin plane