"Hypercomplex Numbers in geometry and Physics" 2 (12), Vol 6, 2009
j012

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

A Special Class of Finsler Geometries and de Sitter Spaces
2009jbz | D.G. Pavlov, G.I. Garas'ko, M.L. Fil'chenkov  // geom2004@mail.ru, gri9z@mail.ru, fmichael@mail.ru

Extensions of General Relativity (GR) have been considered. Reasons for generalizing GR related to difficulties of the theory itself as well as a necessity of interpreting the new astronomical observations are indicated. Numerous attempts of generalizing GR being beyond the scope of Riemannian geometry are listed.
A class of spaces conformally coupled to flat Finsler spaces is shown to be singled out among all Finsler spaces. Its dilatation-contraction coefficient and the world function, in terms of which it is expressed, depend only on an interval of the initial flat space. Then from the Finsler geometry self-sufficiency principle it follows that the dilatation-contraction coefficient is a constant divided by the interval, and the world function is a product of a constant and a logarithm of the dilatation-contraction coefficient. Each element of the class possesses an isometric symmetry group, which includes that of the initial flat Finsler space as a proper subgroup, and possesses a conformal symmetry group coinciding with that of initial flat space. If one takes Minkowski space as an initial one, then the above class space is a pseudo-Riemannian space in the four-dimensional region, where the interval in some approximation is changeable by a temporal coordinate, coinciding with de Sitter space in the same approximation.

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On the Relationship Between Anisotropic Riemannian Metrics and Finsler Ones
2009jby | M.L. Fil'chenkov, Yu.P. Laptev  // fmichael@mail.ru

A possibility of representing Berwald-Moor type Finsler metrics as a product of two anisotropic Riemannian metrics has been considered. If spatial determinants of the Riemannian metrics vanish, then the factorization reduces space dimension. Nonzero determinants exist only in a limited interval of the Riemannian metric anisotropy parameters corresponding to complex coefficients of the Finsler metrics.

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Polynomial metrics, match processes and K-ingles
2009jbx | A.V. Koganov  // Institute of system researches of RAS, Moscow, Russia, koganow@niisi.msk.ru

It showed that any Finsler metric of polynomial type in linear spaces correspond the process which defined by particle derivates and had match group of invariance. It inputted the notion of polynomial generalization for metrics of Galileo, Euclid and Minkovsky types on base of standard connection. It showed that the special geometrical K-placed relations (K-ingle) may be inputting on base of polynomial metrics in space of any dimension and any number K. The standard norms and scalar products are particle cases for 1- and 2-ingle. It had the natural operation of decreasing number of places in ingle.

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Wagner's Generalized Groups and their Applications in Geometry and Physics
2009jbw | V.G. Zhotikov  // Moscow Institute for Physics and Technology, Russia, Tomsk State Pedagogical University, Russia, Zhotikov@yandex.ru

Properties important for applications in geometry and physics of a special class of the semigroups are described: Wagner's so-called the generalized groups. The last are known in the foreign literature still as the inverse semigroups. Questions of introduction the theory of the generalized groups and generalized grouds in the physics, resulting to the new, more general the laws of preservation and the predictions of the new physical phenomena are discussed.

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Lagrange aproach in (n+1)-dimensional "Einstein-Gauss-Bonnet" model and n-dimensional Berwald-Moor metric
2009jbv | V.D. Ivashchuk  // Center for Gravitation and Fundamental Metrology, VNIIMS, Moscow, Russia, Institute of Gravitation and Cosmology, Peoples Friendship University, Moscow, Russia, ivashchuk@mail.ru

A $(n +1)$-dimensional Einstein-Gauss-Bonnet (EGB) model is considered. For diagonal cosmological metrics the equations of motion are written as a set of Lagrange equations with the Lagrangian containing two ``minisuperspace'' metrics on $\R^{n}$: a 2-metric of pseudo-Euclidean signature and Finslerian 4-metric that is proportional to $n$-dimensional Berwald-Moor 4-metric. For the case of synchronous time variable the equations of motion reduce to an autonomous system of first order differential equations. For the case of the ``pure'' Gauss-Bonnet model exact solutions with power-law and exponential dependence of scale factors (w.r.t. synchronous time variable) are presented. In the case of EGB cosmology, it is shown that for any non-trivial solution with exponential dependence of scale factors $a_i(\tau) = A_i \exp( v^i \tau)$ there are no more than three different numbers among $v^1,...,v^n$.

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Finslerian N-Spinors in the Relational Approach
2009jbu | S.V. Bolokhov   // bol-rgs@yandex.ru

A close connection is established between the special class of mathematical objects called Finslerian N-spinors and the apparatus of the relational model of space-time. Some physical applications of the Finslerian spinors formalism are shown in the context of the relational approach in physics.

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On Cartan Spaces with the m-th Root Metric $K(x,p)= \sqrt[m] {a^{i_{1}i_{2}...i_{m}}(x)p_{i_{1}}p_{i_{2}}...p_{i_{m}}}$
2009jbt | Ch. Atanasiu, M. Neagu  // Faculty of Mathematics and Informatics, University "Transilvania" of Bra\c{s}ov, Romania, gh_atanasiu@yahoo.com, mircea.neagu@unitbv.ro

The aim of this paper is to expose some geometrical properties of the locally Minkowski-Cartan space with the Berwald-Moor metric of momenta $L(p)=\sqrt[n]{p_{1}p_{2}...p_{n}}$. This space is regarded as a particular case of the $m$-th root Cartan space. Thus, Section 2 studies the v-covariant derivation components of the $m$-th root Cartan space. Section 3 computes the $v$-curvature d-tensor $S^{hijk}$ of the $m$-th root Cartan space and studies conditions for S3-likeness. Section 4 computes the T-tensor $T^{hijk}$ of the $m$-th root Cartan space. Section 5 particularizes the preceding geometrical results for the Berwald-Moor metric of momenta.

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Dynamics in $D\geq 2$-order Phase Space in the Basis of Multicomplex Algebra
2009jbs | R.M. Yamaleev  // Universidad Nacional Autonoma de Mexico, Mexico, Joint Institute for Nuclear Research, Dubna, Russia, iamaleev@servidor.unam.mx

We use commutative {\it algebra of multicomplex numbers}, to construct oscillator model for Hamilton-Nambu dynamics. We propose a new dynamical principle from which it follows two kind of Hamilton-Nambu equations in $D\geq 2$-dimensional phase space. The first one is formulated with $(D-1)$-evolution parameter and a single Hamiltonian. The Hamiltonian of the oscillator model in a such dynamics is given by $D$-degree homogeneous form. In the second formulation, vice versa, the evolution of the system along a single evolution parameter is generated by $(D-1)$ Hamiltonian. The latter is given by Nambu equations in $D\geq 3$-dimensional phase.

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Idempotents and Nilpotents in the Clifford Algebra of Euclidean 3-Space and Their Interconnections in Physics
2009jbr | O. Mornev  // mornev@mail.ru

Within the framework of Space Algebra, the Clifford algebra $Cl_{3} $ generated by the three-dimensional Euclidean space $E_{3} $ over ${\rm {\mathbb R}}$, a structure of idempotents and nilpotents of index 2 is investigated. The general view of these elements is derived \textit{ab init}, and their algebraic and geometric properties are revealed. The equivalence of action of the groups of phase transformations $(U_{1} )$ and rotations $(SO_{3} )$ on the nilpotents of index 2 is discovered: the phase transformations of the nilpotent, which are realized by its multiplications on the complex exponents, lead to spatial rotations of the nilpotent in $E_{3} $, and vice versa. It is proved that the nilpotents of index 2 are the unique elements of $Cl_{3} $, for which the equivalence of action of the groups $U_{1} $ and $SO_{3} $ takes place; thus, this property of nilpotents is a characteristic one. The results obtained are applied to analyzing geometry of vacuum solutions to the Maxwell equations without sources, which describe plane harmonic electromagnetic waves, the photons, with two types of helicity. On the basis of the analysis performed, the non-formal hypothesis is formulated that the real physical space is at least a six-dimensional one: in the minimal case its basis consists of six linearly independent elements, three vectors and three bivectors generated by these base vectors.

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Octonions and Moving Equations of Probabilities
2009jbq | G.A. Quznetsov  // Chelyabinsk State University, Russia, gunn@mail.ru, quznets@yahoo.com

The expression of pointlike event probabilities in terms of octonions gives moving equations, which are similar to Dirac's equations.

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Gravitation law and source model in the anisotropic geometrodynamics
2009jbp | S.V. Siparov  // State University of Civil Aviation, St. Petersburg, Russia, sergey@siparov.ru

The GRT modification taking into account the dependence of metric on the velocities of the sources is built. It is shown that this dependence follows from the equivalence principle and from the inseparability of the field equations and geodesics equations. As it is known, the latter are the conditions of the field equations solvability, and their form coincides with Newtonian one only in the lowest approximation. The obtained modification provides the explanation for the flat character of the rotation curves of spiral galaxies, for Tully-Fisher law, for some specific features of globular clusters behavior and for the essential excess of the observable gravitational lens effect over the predicted one. Neither dark matter nor arbitrary change of dynamics equations appeared to be needed. Important cosmological consequences are obtained.

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About Shape of Julia Sets Analogous on Double Numbers Plane
2009jbo | Pavlov D.G., Panchelyuga M.S., Panchelyuga V.A.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, Institute of Theoretical and Experimental Biophysics of RAS, Pushchino, Russia, panvic333@yahoo.com

Double-numbers analogous of Julia sets pre-fractals in the case of iteration of $z_{n+1} \to z_{n}^{2} +c,$ for $c\neq0$ are constructed. Numerical algorithm, which allows correct visualization of the Julia set pre-fractals is described and limits of it applicability in the case of $c=0$ are illustrated. Analytical methods, which allow studying of the Julia sets shapes are described and application of the methods to pre-fractals of low orders is demonstrated.

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Fundamentals of Elementary Relations Theory
2009jbn | V. A. Panchelyuga  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, Institute of Theoretical and Experimental Biophysics of RAS, Pushchino, Russia, panvic333@yahoo.com

Main goal of the paper is maybe not so much to present our results on the fundamentals of elementary relations theory as rather than draw attention to a problem, which steel unstated despite the fact that it implicitly exists in many logical, mathematical and physical models. This is the problem of elementary, i.e. indecomposable to more simple, relations. Due to extremely general nature of the notion of relation it underlies such the most general concepts of contemporary science as, for example, notion of number, symmetry, interaction, space-time, and so on. Because of this, careful investigation of the elementary relations problem can give us a way to understand not only the origin of the above-mentioned notions but also its possible limits.

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