జర్నల్ ఆఫ్ జెనరలైజ్డ్ లై థియరీ అండ్ అప్లికేషన్స్

Found general expressions for infinite series of equations Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, (A), (B) and (C) have a Common Prime Factor. However, these expressions also hold true for fractional and negative values of x, y, and z, and cover the range of numbers from the infinitely small to the infinitely large.

Extending the definition of Lie algebroid from one base manifold to a pair of diffeomorphic base manifolds, we obtain the generalized Lie algebroid. When the diffeomorphisms used are identities, then we obtain the definition of Lie algebroid. We extend the concept of tangent bundle, and the Lie algebroid generalized tangent bundle is obtained. In the particular case of Lie algebroids, a similar Lie algebroid with the prolongation Lie algebroid is obtained. A new point of view over (linear) connections theory of Ehresmann type on a fiber bundle is presented. These connections are characterized by a horizontal distribution of the Lie algebroid generalized tangent bundle. Some basic properties of these generalized connections are investigated. Special attention to the class of linear connections is paid. The recently studied Lie algebroids connections can be recovered as special cases within this more general framework. In particular, all results are similar with the classical results. Formulas of Ricci and Bianchi type and linear connections of Levi-Civita type are presented.

In this work, it is considered that the vector space is composed by the infinitesimal deformations of the model Z3-filiform Lie algebra Ln,m,p. By using these deformations, all the Z3-filiform Lie algebras can be obtained, hence the importance of these deformations. The results obtained in this work, together with those obtained by Khakimdjanov and Navarro (J. Geom. Phys. 2011 and 2012), lead to compute the total dimension of the mentioned space of deformations.