a b s t r a c tLet µ be a given probability measure and M µ the set of µ-equivalent strictly positive probability densities. In this paper we construct a Banach manifold on M µ , modeled on the spacewhere p is a reference density, for the non-parametric q-exponential statistical models (Tsallis's deformed exponential), where 0 < q < 1 is any real number. This family is characterized by the fact that when q → 1, then the non-parametric exponential models are obtained and the manifold constructed by Pistone and Sempi is recovered, up to continuous embeddings on the modeling space. The coordinate mappings of the manifold are given in terms of Csiszár's Φ-divergences; the tangent vectors are identified with the one-dimensional q-exponential models and q-deformations of the score function.

Given an Orlicz function H satisfying the ∆ 2 property at zero, one can use the Orlicz sequence space H to define a tensor norm g c H and the minimal (H c-nuclear) and maximal (H c-integral) operator ideals associated to g c H in the sense of Defant and Floret. The aim of this paper is to characterize H c-integral operators by a factorization theorem.

We obtain the optimal system’s generating operators associated with a generalized Levinson–Smith equation; this one is related to the Liénard equation which is important for physical, mathematical, and engineering points of view. The underlying equation has applications in mechanics and nonlinear dynamics as well. This equation has been widely studied in the qualitative scheme. Here, we treat the equation by using the Lie group method, and we obtain certain operators; using those operators, we characterized all invariants solutions associated with the generalized equation of Levinson Smith considered in this paper. Finally, we classify the Lie algebra associated with the given equation.

Se caracterizan las soluciones invariantes para la ecuación de Chazy a partir de los operadores generadores del álgebra óptima, la cual fue obtenida mediante el grupo de simetrías de Lie correspondiente a dicha ecuación.

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