Leonardo Solanilla

Noether's Theorem on Surfaces

In this article we prove a version of Noethers Theorem (of Calculus of Variations) which is valid for a general regular (compact) surface. As a special feature, the Lie group of transformations is allowed to act on the Cartesian product of the surface and the functional space. Additionally, we apply the Theorem to a problem in Classical Differential Geometry of surfaces. The given application is actually an example showing how Noethers Theorem can be used to construct invariant properties of the solutions to variational problems defined on surfaces, or equivalently, of the solutions to the associated Euler-Lagrange equations resulting from them.

The story of Landen, the hyperbola and the ellipse

We establish a relation among the arc lengths of a hyperbola, a circle and an ellipse.

MEMORIA SOBRE LA EMERGENCIA DE LAS FUNCIONES ELÍPTICAS

Resumen es: En este articulo respondemos algunos interrogantes que noshemos formulado sobre y en torno a la emergencia historicade las funciones elipticas en la prim...

Proyección quincuncial de Peirce

Abstract. We present the essential theoretical basis and prove concrete practical formulas to compute the image of a point on the terrestrial sphere under Peirce quincuncial projection. We also develop a numerical method to implement such formulas in a digital computer and illustrate this method with examples. Then, we briefly discuss the criticism of Pierpont on the correctness of Peirce’s formula for the projection. Finally, we draw some conclusions regarding the generalization of Peirce’s original idea by means of SchwarzChristoffel transformations.

Memory about the elliptic functions's emergence

Resumen es: En este articulo respondemos algunos interrogantes que noshemos formulado sobre y en torno a la emergencia historicade las funciones elipticas en la prim...

Valuación de la influencia local en un modelo lineal generalizado mediante el operador de horma

In this paper we develop an algorithm for assessing the effect of small perturbations of the data on the validity of a postulated generalized linear model. The procedure is based on the geometric notion of shape operator, a single mathematical object that gathers together all the normal curvatures of a given influence graph or hypersurface. In addition to introducing relevant theoretical notions and explaining the foundations of the local influence assessment method in a generalized linear model with a canonical link, we provide a detailed example of application together with the explicit R code implementing the algorithm.

Canonical Euler-Lagrange equations and Jacobi's theorem on regular surfaces

In this article we establish conditions under which canonical variables can be defined for a variational problem defined on a geometric (compact) surface. Also, we show the form the corresponding Euler-Lagrange equations assume once we rewrite them in terms of such canonical variables. Furthermore, we prove a version of Jacobis theorem generalizing the univariate standard version of this theorem. The main results are applied to the conformal Gauss curvature functional.