The Euler characteristic of the symmetric product of a finite CW-complex

In this paper it is first shown that if M is a 2-dimensional surface, then $M^{(m)}$ is orientable if and only if M is orientable. Using the Macdonald\u27s result [5]. the Betti numbers of $M^{(m)}$ are expressed explicitely, the Euler characteristic is found and it depends only on $m$ and the Euler characteristic of M. Moreover, the formula for the Euler characteristic is proved alternatively also without using the Macdonalds\u27s results

A Tensorial Approach to the Description of Molecular Distortions I. Tetrahedral Molecules

The inclusion of a tetrahedral XY4 molecule (or ion) in a crystal is, very often, followed by a lowering of its symmetry. In order to describe the apparent distortions of the tetrahedron, second-rank tensors were constructed. It was shown that the characteristic surface of such a tensor is always an ellipsoid. The relative lengths of the axes of the ellipsoid and their position with respect to the symmetry elements of the XY4 group can be used to determine the effective symmetry of the molecule, as well as the degree of its distortion. Some of the spectral properties of the studied compounds can also be predicted. 36 S04 ions with accurately refined structures were investigated and the results obtained by this method were compared with the results\u27 obtained by other-= methods. A correlation of rather high significance (r2 = = 0.97) was found between the main components of the tensor and the frequencies of the components of the antisymmetric stretching vibration (V3) of the molecule

Description of Molecular Distortions III. Trigonally-Planar XY3 Molecules

Second-rank tensors were used to calculate the degree of distortion for the NO3 ions in a number of crystalline compounds. All NO3 ions appear to be strictly planar, as found in a previous study1. A significant correlation, between the main components of the tensor and the wavenumbers of the components of the antisymmetric stretching vibration (r3) of the NO3 ion, was found

Geometrical Aspects of the Electromagnetic Field

The space–time is based on space as three-dimensional sphere, space rotations also as three-dimensional sphere and time which is homeomorphic to the Euclidean three-dimensional space. There are four basic exchanges among them and four induced exchanges, which lead to the basic interactions in the nature. This geometrical approach enables to obtain new viewpoint especially on the basic interactions and their geometrical interpretations. More attention is devoted to the electromagnetic interactions, where the magnetic field of the spinning bodies is studied separately from the electromagnetic field of the charged bodies. It is also considered the gravitational interaction in order to emphasize the similarities between them and the properties which separate them

Research of Gravitation in Flat Minkowski Space

In this paper it is introduced and studied an alternative theory of gravitation in flat Minkowski space. Using an antisymmetric tensor, which is analogous to the tensor of electromagnetic field, a non-linear connection is introduced. It is very convenient for studying the perihelion/periastron shift, deflection of the light rays near the Sun and the frame dragging together with geodetic precession, i.e. effects where angles are involved. Although the corresponding results are obtained in rather different way, they are the same as in the General Relativity. The results about the barycenter of two bodies are also the same as in the General Relativity. Comparing the derived equations of motion for the $n$-body problem with the Einstein-Infeld-Hoffmann equations, it is found that they differ from the EIH equations by Lorentz invariant terms of order $c^{-2}$.Comment: 28 page

Deformation of the Planetary Orbits Caused by the Time Dependent Gravitational Potential in the Universe

In the paper are studied the deformations of the planetary orbits caused by the time dependent gravitational potential in the universe. It is shown that the orbits are not axially symmetric and the time dependent potential does not cause perihelion precession. It is found a simple formula for the change of the orbit period caused by the time dependent gravitational potential and it is tested for two binary pulsars.Comment: 7 page

New approach to the fractional derivatives

We introduce a new approach to the fractional derivatives of the analytical functions using the Taylor series of the functions. In order to calculate the fractional derivatives of f, it is not sufficient to know the Taylor expansion of f, but we should also know the constants of all consecutive integrations of f. For example, any fractional derivative of ex is ex only if we assume that the nth consecutive integral of ex is ex for each positive integer n. The method of calculating the fractional derivatives very often requires a summation of divergent series, and thus, in this note, we first introduce a method of such summation of series via analytical continuation of functions