finsler geometry, hypercomplex numbers and physics
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Articles about polynumbers and Berwald-Moor metrics

(Polynumbers are the commutative and associative hypercomplex numbers)

On the possibility of the realization of a tringle in a 3D space with a scalar product
2009jaz | D.G. Pavlov, G.I. Garas  //  , 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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11-01.pdf, 678,728 Kb, PDF

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

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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11-02.pdf, 604,586 Kb, PDF

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

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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11-03.pdf, 672,286 Kb, PDF

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,

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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11-04.pdf, 1291,72 Kb, PDF

Configuratrix and resultant
2009jav | N.S. Perminov  // Kasan State University, Kazan, Russia,

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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11-05.pdf, 576,836 Kb, PDF

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,

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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11-10.pdf, 1600,999 Kb, PDF

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,

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 } = 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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11-11.pdf, 1744,942 Kb, PDF

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

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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11-14.pdf, 604,950 Kb, PDF


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10-01.pdf, 763,712 Kb, PDF


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10-02.pdf, 744,146 Kb, PDF


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10-03.pdf, 767,26 Kb, PDF

Spectral properties and applications of the numerical multilinear algebra of m-root structures
2008jbr | V. Balan  // University Politehnica of Bucharest, Faculty of Applied Sciences;

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.

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10-09.pdf, 603,396 Kb, PDF


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10-10.pdf, 518,384 Kb, PDF


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10-12.pdf, 543,342 Kb, PDF


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09-01.pdf, 537,749 Kb, PDF


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09-02.pdf, 492,197 Kb, PDF


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09-03.pdf, 789,270 Kb, PDF


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09-07.pdf, 532,765 Kb, PDF

Poly-Numbers (Matrions) in Biological and Computer Informatics
2008jas | Sergey Petoukhov, Elena Petoukhova  // Mechanical Engineering Research Institute RAS, Moscow,

The article is devoted to $2^n$-dimensional poly-numbers, which generalize complex and double numbers on the basis of a block-fractal (or Kronecker) algorithm. These poly-numbers were named circular and hyperbolic matrions correspondingly. They were constructed in a course of investigations of genetic code systems from the viewpoint of matrix methods of informatics. Data about algebras of these poly-numbers are presented. A meaning of these poly-numbers for theoretical biology and information science is under discussion.

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09-08.pdf, 1502,55 Kb, PDF


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08-01.pdf, 276,402 Kb, PDF


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08-02.pdf, 293,326 Kb, PDF


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08-03.pdf, 323,160 Kb, PDF

Einstein Equations for the Homogeneous Finsler Prolongation to TM, with Berwald-Moor Metric
2007jbv | Atanasiu Gh., Brinzei N.  // "Transilvania" University, Brasov, Romania,,

Within the geometrical framework provided by (h,v)-metric structures, an important case is that of the homogeneous prolongation (lift) of a Finsler metric to the tangent bundle TM, constructed by R. Miron. In this case, we perform a study of Einstein equations. A special attention is paid to the Berwald-Moor metric, and to metrics conformally related to it.

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08-05.pdf, 299,891 Kb, PDF

On the World function and the relation between geometries
2006jaz | Garas`ko G. I.  // Electrotechnical Institute of Russia, Moscow,}

It is shown that the World function can be regarded as a link between the qualitatively different geometries with one and the same congruence of the world lines (geodesics). If the space in which the World function is defined is a polynumber space, then the hypothesis of the analyticity of the vector field of the generalized velocities of the world lines lead to the strict limitations on the structure of the World function. Main result: Minkowskian space and polynumber space correspond to the same physical World.

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wf-gar.pdf, 172,805 Kb, PDF 05-01.pdf, 672,999 Kb, PDF

2005jbx | Pavlov D. G.

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04-03.pdf, 272,312 Kb, PDF


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04-04.pdf, 224,566 Kb, PDF

The generalized Finslerian metric tensors
2005jbu | Lebedev S. V.  // Baumann University's Institute of Applied Math@Mech

The generalized Finslerian metric tensors are proposed. These tensors can have different number of indeces dependent on space dimension as well as space properties. The relationship of these tensors with the Finsler spaces associated with commutative associative algebras is analyzed. Nearest perspectives to research of the tensors of this type are discussed. The generalized differential equations of Finsler geodesics are derived and discussed.

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04-06.pdf, 223,822 Kb, PDF

The notions of distance and velocity modulus in the linear Finsler spaces
2005jaz | Garas'ko G. I., Pavlov D. G.

The formulas for the 3-dimensional distance and the velocity modulus in the 4-dimensional linear space with the Berwald-Moor metrics are obtained. The used algorithm is applicable both for the Minkowski space and for the arbitrary poly-linear Finsler space in which the time-like component could be chosen. The constructed here modulus of the 3-dimensional velocity in the space with the Berwald-Moor metrics coincides with the corresponding expression in the Galilean space at small (non-relativistic) velocities, while at maximal velocities, i.e. for the world lines lying on the surface of the cone of future, this modulus is equal to unity. To construct the 3-dimensional distance, the notion of the surface of the relative simultaneity is used which is analogous to the corresponding speculations in special relativity. The formulas for the velocity transformation when one pass from one inertial frame to another are obtained. In case when both velocities are directed along one of the three selected straight lines, the obtained relations coincide with the analogous relations of special relativity, but they differ in other cases. Besides, the expressions for the transformations that play the same role as Lorentz transformations in the Minkowski space are obtained. It was found that if the three space coordinate axis are straight lines along which the velocities are added as in special relativity, then taking the velocity of the new inertial frame collinear to the one of these coordinate axis, one can see that the transformation of this coordinate and time coordinate coincide with Lorentz transformations, while the transformations of the two transversal coordinates differ from the corresponding Lorentz transformations.

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03-01-e.pdf, 418,171 Kb, PDF 03-01.pdf, 692,959 Kb, PDF


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03-02.pdf, 462,948 Kb, PDF

The normal conjugation on the poly-number set.
2004jby | Garasko G.I., Pavlov D.G.

The poly-number set is an example of linear space with several poly-linear forms. The concept of normal conjunction is introduced on the set of non-degenerated n-numbers. The normal conjunction is a (n-1)-nary operation. It is commutative for each argument, but generally not associative. For complex and hyperbolic numbers the generalized conjunction is equivalent to usual one. The normal conjunction may be applied for scrutiny of algebraic and geometric structures of n-number coordinate spaces. It is also useful for introducing such concepts like scalar product and angular characteristics of two and more numbers (vectors)

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02-02.pdf, 613,179 Kb, PDF

Generalized-analytical functions and the congruence of geodetic.
2004jbx | Garas'ko G. I.

Some properties of generalized-analytical functions of poly-number variable are being studied in this job. We can confront many spaces of affine connectedness with the $\{f^i;\Gamma^{i}_{kj}\}$ class of such functions. In each space the congruence of geodetic associated with the given class of general-analytic functions is defined. If the vector field $f^i$ is tangent to one of the geodetic of congruence in each point of space there are certain restrictions for the generalized-analytical function itself.

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02-03.pdf, 502,768 Kb, PDF

Generalization jf Scalar Product Axioms
2004jay | Pavlov D. G.

The concept of scalar product is vital in studying basic properties of either Euclidean or pseudo-Euclidean spaces. A generalizing of a special sub-class of Finslerian spaces, that we will call the polylinear, is presented in the work. The idea of scalar polyproduct and of related fundamental metric polyform has been introduced axiomatically. The definition of different metric parameters such as the vector length and the angle between vectors are founded on the idea. The concept of orthogonallity is also generalized. Some peculiarities of the geometry of the four-dimensional linear Finslerian space related to the algebra of commutative-associative hypercomplex numbers, that are called Quadranumerical, are proved in the concrete polyform.

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01-02-e.pdf, 252,553 Kb, PDF 01-02.pdf, 578,823 Kb, PDF

Chronometry of the three-dimensional time
2004jax | Pavlov D. G.

The concept of the multi-dimensional time has tried not once to take its place in natural science, but every time under the pressure of some paradox was rejected. Meanwhile a philosophical question: why the space admits quite a number of dimensions and the time dos not, still preserves. In this work a new attempt has been made to resolve the matter, by switching from the traditional quadratic metrics to the Finslerian one, which may admit an arbitrary degree of the vector component that is included into the metric function. Though the offered method enables us to build continuums of time of any natural dimensionality, in order to demonstrate the specificity of the raised topic this study will focus on a simple (after rather trivial two-dimensional case) example of three temporial dimensions.

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01-03-e.pdf, 353,98 Kb, PDF 01-03.pdf, 655,958 Kb, PDF

Four-dimensional time
2004jaw | Pavlov D. G.

The generalized metric space, that can be called the flat four-dimensional time, is based on the Berwald-Moore's Finslerianview of metric function. This variety let us introduce physical notions: the event, the world lines, the reference frames, the multitude of relatively simultaneous events, the proper time, the three-dimensional distance, the speed, etc. It is demonstrated how from the point of the physical observer, associated with the world line, in absolutely symmetrical four-dimensional time the contraposition of the coordinate takes place, that defines its proper time, with the ones that appear as the result of the measurements made with the help of sample signals. When the signals correspond with lines, which are practically parallel to the world line of the observer, he starts to see the three-dimensional space which at the limit is the Euclidean space.

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01-04-e.pdf, 241,19 Kb, PDF 01-04.pdf, 601,258 Kb, PDF

Properties of spaces associated with commutative-associative H3 and H4 algebras
2004jau | Lebedev S. V.

In the first part of this work a real axis of the space associated with the H3 algebra and the lines parallel to this axis are interpreted as the world lines of resting particles; surface of simultaneity is used for introduction of a distance between the real axis and a line parallel thereto. The coordinate system similar to a polar one can be introduced on this surface such that this allows us to reveal its simplest invariant transformations. In the second part of this paper the Lorentz transformations in form of special kind of rotations in the space associated with H4 algebra are presented.

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01-06-e.pdf, 300,840 Kb, PDF 01-06.pdf, 617,769 Kb, PDF

Generalized-analytical functions of poly-number variable
2004jat | Garas`ko G. I.

We introduce the notion of the generalized-analytical function of the poly--number variable, which is a non--trivial generalization of the notion of analytical function of the complex variable and, therefore, may turn out to be fundamental in theoretical physical constructions. As an example we consider in detail the associative-commutative hypercomplex numbers H4 and an interesting class of corresponding functions.

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01-07-e.pdf, 253,152 Kb, PDF 01-07.pdf, 535,308 Kb, PDF

Three-numbers, which cube of norm is nondegenerate three-form
2004jap | Garas`ko G. I.

Arbitrary three-form can be put in a canonical form. The requirement of existence of two-parametric Abelian Lie group to play the role of group of symmetry for three-form admits selecting the three-forms that correspond to three-numbers and finding all the three-numbers which cube of norm is a non-degenerate three-form with respect to a special coordinate system. There are exactly two (up to isomorphism) such sets of hypercomplex numbers, namely the sets: C3, H3. They can be regarded as generalizations of complex and binary (hyperbolic) bi-numbers to the case of three-numbers.

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01-11-e.pdf, 216,896 Kb, PDF 01-11.pdf, 451,575 Kb, PDF

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