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.
In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in the cases of both symmetric and non-symmetric connection. Although the curvature-flatness idea is classic for symmetric connection, our main theorems about flatness solutions are completely new, leaving for a while the point of view of differential geometry and entering that of PDEs. The second goal is to introduce and study some strong partial differential relations associated to curvature-flatness. The third goal is to introduce and analyze some vector spaces of exotic objects that change the meaning of a generalized Kronecker delta projection operator, in order to discover new PDEs implying curvature-flatness. Significant examples clarify some ideas.
The theory of statistical manifolds w.r.t. a conformal structure is reviewed in a creative manner and developed. By analogy, the γ-manifolds are introduced. New conformal invariant tools are defined. A necessary condition for the f-conformal equivalence of γ-manifolds is found, extending that for the α-conformal equivalence for statistical manifolds. Certain examples of these new defined geometrical objects are given in the theory of Iinformation.
The paper studies the Lie symmetries of the nonlinear Fokker–Planck equation in one dimension, which are associated to the weighted Kaniadakis entropy. In particular, the Lie symmetries of the nonlinear diffusive equation, associated to the weighted Kaniadakis entropy, are found. The MaxEnt problem associated to the weighted Kaniadakis entropy is given a complete solution, together with the thermodynamic relations which extend the known ones from the non-weighted case. Several different, but related, arguments point out a subtle dichotomous behavior of the Kaniadakis constant k, distinguishing between the cases k∈(−1,1) and k=±1. By comparison, the Lie symmetries of the NFPEs based on Tsallis q-entropies point out six “exceptional” cases, for: q=12, q=32, q=43, q=73, q=2 and q=3.
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