The aim of this paper is to develop on the 1-jet space the Finsler-like geometry (in the sense of distinguished (d-) connection, d-torsions and d-curvatures) for a 1-parameter deformation of the Berwald-Mo´or metric of order three. Some field-like geometrical theories (gravitational-like and electromagnetic-like) produced by our 1-parameter deformation of the Berwald-Mo´or metric are also exposed.

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Identically solvable finsler geometries 2010jdw | Garasko G.I. // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, gri9z@mail.ru

We suggest an algorithm to search for the identically solvable Finsler geometries which provides the possibility to find some solvable Finsler geometries that are not identically solvable. This algorithm is closely related to the reflection on the space whose dimension is one unit less than the dimension of the Finsler space on itself. This reflection must coincide to its own reverse and possess some other properties. For the spaces of arbitrary dimension, the identical reflection corresponds to the Euclidean space, the reflection with the simultaneous change of the sign of all coordinates - to the pseudo Euclidean space, and the reflection with the inversion of all the coordinates corresponds to the space with Berwald-Moor metric.

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On n-ary subgroups of a special n-ary group 2010jcz | Galmak A.M., Vorobev G.N., Balan V.D. // Mogilev State University of Food Technology, Mogilev, Belarus; Polytechnical University of Bucharest, Bucharest, Romania, mgup@mogilev.by, vbalan@mathem.pub.ro

We consider the Cartesian power An−1 (n ≥ 3) of the group A, which admits a subgroup B such that the factor group A/B is cyclic and with its order dividing n − 1. On An−1 we construct an n−ary group structure < An−1, [ ]n,n−1 > with its operation similar to the one defined by E. Post for n−ary permutations. This structure is shown to admit semi-invariant - but generally not invariant, n−ary subgroups.

Obtained was an analog for Cauchy's formula for non-degenerate commutative-associative hypercomplex numbers (polynumbers), including algebra of complex numbers or direct summ of complex algebras as means of subalgebra. Therewith manifested are the reasons why Cauchy's formula in polynumbers Hn which are the direct sums of nothing but actual algebras is so hard to obtain.

Using double numbers algebra we develop algebraic version of 2D relativity theory, which takes intermediate place between special and general relativity. In space-time free of matter the main object of the theory - hyperbolic potential F - is h-holomorphic function of double variable. Physically it is responsible for local splitting of space-time onto time and space directions in conformally deformed flat Minkowski space. It is shown that the effect of conformal deformation is in principle observable with the help of experiments concerning measurements with spatially separated clocks. Space-time with matter is described by relation F,h ≠ 0. The dynamics of hyperbolic field is described by special action, depending only on hyperbolic square of non-holomorphicity. It is shown, that field equation are non-linear h-conjugated wave equations with self-action. Specific properties of these equations are: 1) presence of the first integral; 2) compatibility (integrability) condition, defining class of admissible fields G(H2). The latter condition can be viewed as generalization of hyperbolic Cauchy-Riemannian condition and it is crucial for construction of consistent and reliable physical theory of space-time and matter in 2D. As an example we consider static 2D universe with 1D deformed bar. Some aspects of relations of SR and GR to the approach are discussed. We formulate super-extremum principle, allowing one to calculate fundamental constants and boundary conditions of the theory.

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Double numbers 2010jbw | Pavlov D.G., Garasko G.I. // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, geom2004@mail.ru, gri9z@mail.ru

There is an attempt to show that there is much more in common between the complex numbers and the double (hyperbolically complex) numbers than it is usually thought to be. With this the new and non-trivial properties of the analytical functions of the double number variable are discovered. For example, there is a relation between these functions and the hyperbolic potential and solenoidal vector fields on the pseudo Euclidean plane. Besides, it is shown how many structures on the complex plane can be one-to-one mapped at their hyperbolical analogues. This repudiates the magic properties of the complex numbers, and particularly, leads to the understanding that the analytical functions of complex numbers can be given by two scalar functions not of two but of one real variable each.

On the basis of the analogy of complex numbers analytical functions with two-dimensional electro- and magnetostatic fields there was made an assumption considering the existence of such correspondence between h-analytical functions of the binary variable and some other pair of binary physical fields in reality, one of which is hyperbolical source field and another is hyperbolically vortex field. Unlike electro- and magnetostatic fields, this pair is not realized in space but rather in space-time; thus, the sources of the first field are events while force lines of the second vortex constituent are hyperbolas. Essential feature of this hypothetical pair of fields is that it is feasible only in two-dimensional pseudo-Euclidian space and that it is fundamentally incompatible with the Minkowski idea of 4-dimensional space-time. Partially this is the very reason why such fields weren't considered potentially feasible by physicians even in theory. Their immediate discovery is hampered by experimentalists' being used to space-boundary conditions, while they had better work with space-time ones here. Although this pair is incompatible with Minkowskyi space, still it can possibly be realized in 4-dimesional space possessing, in particular, Berwald-Moor Finsler metric function, its discovery in reality, thus, serving a valid reason to substitute the quadratic metric idea of space-time geometry for Finsler one, connected with quartic form.

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.

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.

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.

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.

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 $v^1,...,v^n$.

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.

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.

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.

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.

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.

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.

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.

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 H_{3} 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.

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.

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.

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.

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.

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.

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 ñ} = 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.

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.

Berwald-Moor space $H_{4}$ 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 $H_{4}$ 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)

As a continuation of the ideas in our last work, we determine a new solution
for Einstein equations in vacuum for linearly approximable anisotropic
perturbations of flat Minkowski and Berwald-Moor Finslerian metric. Also, we
determine the effective expressions for geodesics and eikonal for small
anisotropic perturbations of Minkowski and Berwald-Moor metrics and the
changes of the OMPR conditions for the two models. This could in principle
provide the possibility to study the anisotropic properties of space-time in
our galaxy.

In the framework of supersymmetric tensors and multivariate homogeneous polynomials, the talk discusses the 4-th order Berwald-Moor case. The eigenvalues and eigenvectors are determined; the recession and degeneracy vectors, characterization points, rank, asymptotic rays, base index, are studied. As well, the best rank-one approximation is derived, relations to the Berwald-Moor poly-angles are pointed out, and a brief outlook on real-world applications is provided.