Weighted Relative Group Entropies and Associated Fisher Metrics

A large family of new α-weighted group entropy functionals is defined and associated Fisher-like metrics are considered. All these notions are well-suited semi-Riemannian tools for the geometrization of entropy-related statistical models, where they may act as sensitive controlling invariants. The main result of the paper establishes a link between such a metric and a canonical one. A sufficient condition is found, in order that the two metrics be conformal (or homothetic). In particular, we recover a recent result, established for α=1 and for non-weighted relative group entropies. Our conformality condition is “universal”, in the sense that it does not depend on the group exponential.

An expansion of bivariate spline functions

Let Δ denote a triangulation of a planar polygon Ω. For any positive integer 0 ≤ r < k, let denote the vector space of functions in Cr whose restrictions to each triangle of Δ are polynomials of total degree at most k. Such spaces, called bivariate spline spaces, have many applications in surface fitting, scattered data interpolation, function approximation and numerical solutions of partial differential equations. An important problem is to give the function expression. In this paper, we prove that, if (Δ, Ω) is type-X, then any bivariate spline function in can be expressed by a series of univariate polynomials and a special bivariate finite element function in satisfying a so-called integral conformality condition system. We also give a direct sum decomposition of the space . In addition, the dimension of for a kind of triangulation has been determined.