arXiv subjects
Publications and source records attributed to Kostadin Trenčevski.
In this paper are examined general classes of linear and non-linear analytical systems of partial differential equations. Indeed the integrability conditions are found and if they are satisfied, the solutions are given as functional series in a neighborhood of a given point (x=0).
The aim of the present paper is to give a formula for the $k$-th covariant derivative of tensor field along a given curve. In order to do that, first the symbols $P^{i }_{j}$ and $Q^{i }_{j}$ which depend on the Christoffel symbols are introduced. Some properties of them are also given. The main result is given by (3.1) and further it is generalized for $k\in R$.
In this paper is defined an $n$-inner product of type $\langle {\bf a}_1,\cdots ,{\bf a}_n\vert {\bf b}_1\cdots {\bf b}_n\rangle $ where ${\bf a}_1,\cdots ,{\bf a}_n$, ${\bf b}_1, \cdots ,{\bf b}_n$ are vectors from a vector space $V$. This definition generalizes the definition of Misiak of $n$-inner product \cite{2}, such that in special case if we consider only such pairs of sets $\{ {\bf a}_1,\cdots ,{\bf a}_1\}$ and $\{ {\bf b}_1\cdots {\bf b}_n\}$ which differ for at most one vector, we obtain the definition of Misiak. The Cauchy-Schwarz inequality for this general type of $n$-inner product is proved and some applications are given.
In this paper we consider equations of motion for 2-body problem according to an observer close to one of the gravitational bodies. The influence of the Thomas precession of the observer's frame has an important role. The equations of motion are based on a nonlinear connection modified by the action of an orthogonal tensor which synchronizes the 4-velocities of the considered two bodies. Finally we present periastron shift according to an observer inside the gravitational field, using the orthonormal coordinates. This is different approach than that used in General Relativity where the periastron shift is only given as observed far from the massive bodies.