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 "Hypercomplex Numbers in geometry and Physics" 2 (23), Vol 12, 2015
j023
Content of Issue
 BIWAVE EQUATION WITH VECTOR STRUCTURAL COEFFICIENT AND ITS GENERALIZED SOLUTIONS
2015jev | L.A. Alexeyeva // Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science, Republic of Kazakhstan, Almaty, Kazakhstan, alexeeva@math.kz
The biquaternionic wave equation with vector structural coefficient is investigated . With
use of the theory the generalized functions and scalar potentials the fundamental and
generalized solutions of this equation are constructed at any regular or singular right part. The wave and energy properties of elementary decisions are investigated.
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TIME AS A FIELD OF SYNCHRONISED PROPER TIMES OF A NORMAL CONGRUENCE OF WORLD LINES
2015jgv | G.I. Garas’ko // Electrotechnical Institute of Russia, Moscow, Russia, gri9z@yandex.ru, gri9z.wordpress.com The author suggests to interpret time as a field of synchronized proper times of a normal
congruence of world lines. He lays down the conditions allowing to consider a function of a point of the events space as a parameter of evolution. The length of a world line from a normal congruence of extremums is the best suited parameter of evolution, that is, an operation is the same as a point function in the Newtonian mechanics. Whether the world field (the field of the world function or a meatspace at zero-order approximation) and a field of synchronized proper times of a normal congruence of world lines are interrelated? If a field of the proper times can be expressed through a world function only, then hypersurfaces of the world function level are the hypersurfaces of the proper times of normal congruencies of the world lines. A space per se (at any moment of time) is transversal to all the world lines from the normal congruence.
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TIME AS A FIELD OF SYNCHRONIZED PROPER TIMES OF NORMAL CONGRUENCE OF WORLD LINES II
2015jkv | G.I. Garas’ko // Electrotechnical Institute of Russia, Moscow, Russia, gri9z@yandex.ru, gri9z.wordpress.com The paper shows that if one knows a Finsler function ?(p; x), and a function S(x), that
is an operation as a function of coordinates, determining a normal congruence of world
lines, and knows an element of proper times along them, one can draw a differential
equation with partial derivatives for a field of proper times T(x) of the normal congruence
of world lines. Interrelation of such sort between the abovementioned ideas is of special
interest, when hypersurfaces of the level T(x)=const are the transversal supersurfaces to
the normal congruence of the world lines determined by the world function SW(x).
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NULLVECTOR ALGEBRA
2015jlv | S.Ya. Kotkovsky // s_kotkovsky@mail.ru In this paper, we study the properties of biquaternionic divisors of zero («nullquaternions»).
Subalgebra of nullquaternions is closely related to its subclass – «nullvectors», which are complex-valued three-dimension vectors, having zero square. Theorem of nullvector factorization shows that regular nullquaternion can be represented as product of two nullvectors belonging to uniquely defined classes, thus defining the structure of nullquaternion. Theorem of nullvector allelity proves that product of two nullquaternions preserves one of the structure halves of each multiplier. Last circumstance signs for prominent similarity of nullvector algebra with genetics: product of nullvectors is similar to combination of genes in chromosome. We show, that along with nullvectors there exist
«uniform» classes of nullquaternions which are isomorphic to nullvector classes. Regular, uniform nullquaternions and nullvectors represent general classification of nullquaternions
with relation to multiplication.
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 INTEGRAL HYPERBOLIC DYNAMICS OF PARTICLES IN THE MINKOWSKI SPACE-TIME
2015jlv | S.S. Kokarev // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia; RSEC Logos, Yaroslavl, Russia, geom2004@mail.ru, logos-center@mail.ru The paper presents an action of an interacting particles system based on a hyperbolic-spherical-symmetric decision of a wave equation in the Minkowski space-time, space-time analog of the Colomb’s law, and the superposition law. The corresponding motion equations are integro-differential, and their notation according to the Newton’s second law in the relativistic form reveals the dynamic nature of a mass: it becomes a result of a cooperative effect of a part of hyperbolic interaction between the particle and its environment (the Mach’s principle). The author analyses some special cases of the hyperbolic self-action of a solitary world line and interactions between a pair of particles and parallel world lines.
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