"Hypercomplex Numbers in geometry and Physics" 1 (11), Vol 6, 2009
j011

Content of Issue is in the theme. The journal in one file is below.

On the possibility of the realization of a tringle in a 3D space with a scalar product
2009jaz | D.G. Pavlov, G.I. Garas  // НИИ ГСГФ, Bauman Moscow State Technical University, Moscow, Russia, Electrotechnical Institute of Russia, Moscow, Russia

The isometric and conform symmetry groups are of exceptional importance in mathematics and physics that can scarcely be overestimated. The former class of symmetry relates to the invariant of the element of length of the metric space, but the latter class of symmetry relates to the angle invariant. If there exists a continuation of this chain of the symmetry groups, isometric, conform… etc, then there should exist objects tightly connected with this more generic class of symmetry group, which are common to call as tringles or, without any relation to the dimension, as ingles, and, to show the dimension m exceeding 3 -- as m-ingles. It is not possible to have ingles in the Euclidian or pseudo-Euclidian spaces, but, in contrast, it is possible to have ingles in the space with the dimension exceeding 2 and having scalar polyproducts, with the number of the vector arguments also above 2. In the present work, we build a real tringle accurate within a function of one real variable, and we derived its relation to the coordinates of the vectors in the space with a scalar triproduct, where the space is tightly connected with the Bervald-Moor 3D space, which is justified to be called as 3D-time. So, the existence of the tringles, which have been supposed to exist, is rigorously proven that implies a real possibility for m-ingles, with $m3$, to exist.

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Indicatrix volumes of some Finsler spaces of special type
2009jay | G.I. Garas'ko  // HSGPH, Electrotechnical Institute of Russia, Moscow, Russia, gri9z@mail.ru

Indicatrix volumes of some Finsler spaces of special type were obtained. This allows to clarify the question about existence of finite (non-zero) volume element in the Finsler spaces with single time coordinate and in the Finsler spaces with concave indicatrix.

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A self-sufficiency principle in Finsler geometry
2009jax | G.I. Garas'ko  // HSGPH, Electrotechnical Institute of Russia, Moscow, Russia, gri9z@mail.ru

By using the self-suuficiency principle of Finsler geometry, one can derive the field equations, where the gravitational field and electromagnetic field naturally join together as in the pseudo- Riemannian 4D space as well as in the curvilinear Berwald-Moor 4D space; there always exists an energy-momentum tensor related to conservation laws. It has been shown that, in the approximation of small fields, the new geometric approach in the field theory following from the self-sufficiency principle of the Finsler geometry can result in linear field equations valid for several independent fields. When the strength of the fields increases, which means the use of the second approximation, the field equations become generally nonlinear and the fields loose independence that leads to the violation of the superposition principle for each separate field, and results in the interaction among different fields.

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Polyangles and their symmetries in H₃
2009jaw | D.G. Pavlov, S.S. Kokarev  // Research Institute of Hypercomplex Numbers in Geometry and Physics, Friazino, Russia, RSEC "Logos", Yaroslavl, logos-center@mail.ru

We construct bingles and tringles in 3D Berwald-Moor space as additive characteristics of pairs and triples of unit vectors -- lengths and squares on unit sphere (indicatrix). Two kind of bingles (mutual and relative) can be determined analogously to spherical angles $\theta$ and $\varphi$ respectively. We show that mutual bingle is, in fact, norm in space of exponential bingles (bi-space $H_3^{\flat}$), which define exponential representation of polynumbers. It is turned out, that metric of bi-space is the same Berwald-Moor ones. Relative angles are connected with elements of second bi-space $(H_3^{\flat})^{\flat}$ and give possibility for two-fold exponential representation of polynumbers. Apparent formulae for relative bingles and tringles contain non-elementary integrals.

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Configuratrix and resultant
2009jav | N.S. Perminov  // Kasan State University, Kazan, Russia, nikolai-kazan@rambler.ru

In this paper, we obtain an explicit expression for the resultant of $n$ quadratic algebraic equations $\{\partial_{1}S = 0, \ldots, \partial_{n}S=0\}$, where $S$ is a cubic polynomial in $n$ variables, symmetric under permutations of its arguments. Application of this result to the study of Finslerian spaces is discussed.

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Wrap groups of quaternion and octonion fiber bundles
2009jau | S.V. Ludkovsky  // MIREA, sludkowski@mail.ru

This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real R, complex C numbers, the quaternion skew field H and the octonion algebra O. These groups are constructed with mild conditions on fibers. Their examples are given. It is shown, that these groups exist and for differentiable fibers have the infinite dimensional Lie groups structure, that is, they are continuous or differentiable manifolds and the composition $(f,g)\mapsto f^{-1}g$ is continuous or differentiable depending on a class of smoothness of groups. Moreover, it is demonstrated that in the cases of real, complex, quaternion and octonion manifolds these groups have structures of real, complex, quaternion or octonion manifolds respectively. Nevertheless, it is proved that these groups does not necessarily satisfy the Campbell-Hausdorff formula even locally.

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Structure of wrap groups of hypercomplex fiber bundles
2009jat | S.V. Ludkovsky  // MIREA, sludkowski@mail.ru

This article is devoted to the investigation of structure of wrap groups of connected fiber bundles over the fields of real R, complex C numbers, the quaternion skew field H and the octonion algebra O, as well as commutative hypercomplex quadra-algebra. Iterated wrap groups are studied as well. Their smashed products are constructed.

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Types of hypercomplex numbers describing equality, which corresponds to inequalities of Schwarz--Cauchy--Bounjakowsky
2009jas | L.G. Solovey  // lgsolovey@gmail.com

The paper considers a different variants of hypercomplex systems (quasiquaternions) for which inequalities equivalent to Schwarz--Cauchy--Bounjakowsky inequalities exists. The variants are different for systems with complex coefficients but coincide for systems with real coefficients. Paper investigates typical properties of the considered variants.

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Fields analogues of Newton's laws for one model of electro-gravymagnetic field
2009jar | L.A. Alexeyeva  // Institute of mathematics, Alma-Ata, Kazakhstan, alexeeva@math.kz

With use the Hamilton's form of the Maxwell's equations one biquaternion model for electro-gravymagnetic (EGM) field is offered. The equations of the interaction of EGM-fields, generated different charge and current, are built. The field analogues of three Newton's laws are offered for free and interacting charge-currents, as well as total field of interaction. An invariance of the equations at Lorentz transformation is investigated, and, in particular, law of the conservation of the charge-current. It is shown that at fields interaction, this law differs from the well-known one. The new modification of the Maxwell's equations is offered with entering the scalar resistance field in biquaternion of EGM-field tension. Relative formulas of the transformation of density of the masses and charge, current, forces and their powers are built. The solution of the Caushy problem is given for equation of charge-current transformations.

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On fractality of Mandelbrot and Julia sets on double-numbers plane
2009jaq | Pavlov D.G., Panchelyuga M.S., Malykhin V.A., Panchelyuga V.A.  // HSGPH, Institute of Theoretical and Experimental Biophysics RAS, Pushchino, Russia, panvic333@yahoo.com

The paper presents results of numerical calculation of analogues of Mandelbrot and Julia sets on double-numbers plane and for the first time demonstrates their fractal character. Also a short revue of works, which devoted to building of double-numbers Mandelbrot and Julia sets is presented.

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About shape of Julia set at zero parameter on double numbers plane
2009jap | Pavlov D.G., Panchelyuga M.S., Panchelyuga V.A.  // HSGPH, Institute of Theoretical and Experimental Biophysics RAS, Pushchino, Russia, panvic333@yahoo.com

Analytic solution for Julia set on double numbers plane in the case of quadratic map $z_{n+1} \to z_{n}^{2} +c,$ at {\it с} = 0 is presented. Paper illustrates main problems of numerical algorithm creation to calculate the Julia set having correct shape. Despite on simple mathematical character the consideration allows to illustrate main problems of double numbers fractals calculations, which don't exist for complex numbers fractals.

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On fractal structure of space revealing during investigations of local-time effect
2009jao | Victor A. Panchelyuga, Simon E. Shnoll  // HSGPH, Institute of Theoretical and Experimental Biophysics RAS, Pushchino, Russia; Department of Physics, Lomonosov Moscow State University, Moscow, Russia; panvic333@yahoo.com, shnoll@mail.ru

The paper presents experimental investigations of local-time peak splitting right up to second-order splitting. Splitting pattern found in the experiments has a fractal character. A hypothesis about the possibility of high order splitting is proposed. The obtained experimental result leads to a supposition that the real space possess fractal character.

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Search results of preferential direction and the heterogeneity of the Universe on the base of quasars distribution statistic
2009jan | V.Ya. Vargashkin  // Orel State Technical University, Orel, Russia, varg@physics.org

Presents the analysis of histograms of distribution of quasars in the redshift values for the statistical sampling windows, different ways oriented in directions of the celestial sphere. Detected heterogeneity of this distribution having the form of structures of filaments and voids. Global character of anisotropy of distribution of quasars on heavenly sphere is analyzed.

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Minkowski metrics and Berwald-Moor metrics
2009jam | O. Titov  // Geoscience, Australia, olegtitov903@hotmail.com

Berwald-Moor space $H₄$ was proposed by Garas'ko and Pavlov as expansion of Minkowski space. As basic argument allowing such expansion in both geometries was considered presentation of interval like system of isotropic vectors. At the same time, according to statement of authors 'coordinates $(x_0, x_1, x_2, x_3)$ in orthonormal basis of $H₄$ space in non-relativistic approach in geometrical (metrical) sense behave oneself as conventional coordinates of four-dimensional Minkowski space-time'. Present work shows that such statement is incorrect.
(Polemic article)

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