arXiv subjects
Publications and source records attributed to Žarko Mijajlović.
A nonstandard proof of a generalization of Karamata uniform convergence theorem for slowly varying functions is presented. Properties of a related operator $\mathcal{L}$ and its connection with slowly varying functions are discussed.
In this paper we exploit the theory of the dynamical systems to study the dynamics of the standard cosmological model of the universe, which is known as the $Λ$CDM model. We assume that the matter content in our universe consists of barotropic perfect fluids without mutual interaction. Furthermore, we present the appropriate physical interpretation, as well as new dependencies between the scale expansion factor of the universe and cosmological density parameters.
We applied the theory of regularly varying functions to the analysis of the cosmological parameters for the $Λ$CDM model with the matter dominated evolution. Carroll et al. proved in 1992 that for this type of universe with the curvature $k=0, -1$, the expression $H(t)t$\, ($H(t)$ is the Hubble parameter) depends solely on the density parameter $Ω(t)$. Using this result and the theory of regular variation we infer for such universe the complete asymptotics of all main cosmological parameters. More specifically, the following is derived. If the limit $ω= \lim_{t\to\infty} Ω(t)$ does exist and $ω\not= 0$ then the cosmological constant $Λ$ is equal to $0$. If $ω=0$ then for the expansion scale factor $a(t)$ we have $a(t)\sim e^{\sqrt{Λ/3}}$. On the other hand, if the limit $\lim_{t\to\infty} Ω(t)$ does not exist then $a(t)$ bounces between two power functions and therefore has infinitely many flexion points. Hence, the deceleration parameter in this case changes the sign infinitely many times.
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.
In this article a new method of generating sums of like powers is presented.
We discuss some problems in number theory posed by Djuro Kurepa (1907-1993), including his classical left factorial hypothesis that an odd prime $p$ does not divide $0! + 1! + ... + (p-1)!$.