18 research outputs found
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 is orientable if and only if M is orientable.
Using the Macdonald\u27s result [5].
the Betti numbers of are expressed explicitely,
the Euler characteristic is found and it depends only on
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 -body problem with the Einstein-Infeld-Hoffmann equations,
it is found that they differ from the EIH equations by Lorentz invariant terms
of order .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