Our main aim is to apply the theory of regularly varying functions to the asymptotical analysis at infinity of solutions of Friedmann cosmological equations. A new constant Γ is introduced related to the Friedmann cosmological equations. Determining the values of Γ we obtain the asymptotical behavior of the solutions, i.e. of the expansion scale factor a(t) of a universe. The instance Γ < 1 4 is appropriate for both cases, the spatially flat and open universe, and gives a sufficient and necessary condition for the solutions to be regularly varying. This property of Friedmann equations is formulated as the generalized power law principle. From the theory of regular variation it follows that the solutions under usual assumptions include a multiplicative term which is a slowly varying function.

The aim of this article is to exhibit a method for proving that certain analytic functions are not solutions of algebraic differential equations. The method is based on model-theoretic properties of differential fields and properties of certain known transcendental differential functions, as of Γ(x). Furthermore, it also determines differential transcendence of solution of some functional equations.

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We study Hopf algebras over projection functions of the complex vector C X appropriate for computing inversion formulas from discrete mathematics. Using calculus of projection functions introduced in this way, we derived various inversion formulas, including Gould's inversion formula and its generalizations.

In the paper On asymptotic solutions of Friedmann equations (Mijajlovic et al. 2012), the theory of regularly varying functions in the sense of Karamata is applied in an asymptotic analysis of solutions of Friedmann equations. As is well known, solutions of these equations are used to represent cosmological parameters. Therefore, according to the theory of regularly varying functions all cosmological parameters depend on a function ?(t) such that limt?1 ?(t) = 0 and which appears in their integral representation. In this paper we derive a differential equation for the parameter ?(t), discuss its solutions and give some physical interpretations. [Projekat Ministarstva nauke Republike Srbije, br. III 44006]