Articles about polynumbers and BerwaldMoor metrics p999
(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 // ÍÈÈ ÃÑÃÔ, 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 mingles. It is not possible to have ingles in the Euclidian or pseudoEuclidian 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 BervaldMoor 3D space, which is justified to be called as 3Dtime. So, the existence of the tringles, which have been supposed to exist, is rigorously proven that implies a real possibility for mingles, 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 (nonzero) volume element in the Finsler spaces with single time coordinate and
in the Finsler spaces with concave indicatrix.
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A selfsufficiency principle in Finsler geometry 2009jax  G.I. Garas'ko // HSGPH, Electrotechnical Institute of Russia, Moscow, Russia,
gri9z@mail.ru
By using the selfsuuficiency 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 BerwaldMoor 4D space; there always exists an energymomentum 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 selfsufficiency 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,
logoscenter@mail.ru
We construct bingles and tringles in 3D BerwaldMoor 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 (bispace $H_3^{\flat}$), which define exponential representation of polynumbers. It is turned out, that metric of bispace is the same BerwaldMoor ones. Relative angles are connected with elements of second bispace $(H_3^{\flat})^{\flat}$ and give possibility for twofold exponential representation of polynumbers. Apparent formulae for relative bingles and tringles contain nonelementary integrals.
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Configuratrix and resultant 2009jav  N.S. Perminov // Kasan State University, Kazan, Russia, nikolaikazan@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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On fractality of Mandelbrot and Julia sets on doublenumbers 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 doublenumbers plane
and for the first time demonstrates their fractal character. Also a short revue of works, which devoted to building
of doublenumbers 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 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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Minkowski metrics and BerwaldMoor metrics 2009jam  O. Titov // Geoscience, Australia, olegtitov903@hotmail.com
BerwaldMoor 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 nonrelativistic approach in geometrical (metrical) sense behave oneself as conventional coordinates of fourdimensional Minkowski spacetime'. Present work shows that such statement is incorrect.
(Polemic article)
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Spectral properties and applications of the numerical multilinear algebra of mroot structures 2008jbr  V. Balan // University Politehnica of Bucharest, Faculty of Applied Sciences; vbalan@mathem.pub.ro
In the framework of supersymmetric tensors and multivariate homogeneous polynomials, the talk discusses the 4th order BerwaldMoor 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 rankone approximation is derived, relations to the BerwaldMoor polyangles are pointed out, and a brief outlook on realworld applications is provided.
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PolyNumbers (Matrions) in Biological and Computer Informatics 2008jas  Sergey Petoukhov, Elena Petoukhova // Mechanical Engineering Research Institute RAS, Moscow, petoukhov@hotmail.com
The article is devoted to $2^n$dimensional polynumbers, which generalize complex and double numbers on the basis of a blockfractal (or Kronecker) algorithm. These polynumbers 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 polynumbers are presented. A meaning of these polynumbers for theoretical biology and information science is under discussion.
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Einstein Equations for the Homogeneous Finsler Prolongation to TM, with BerwaldMoor Metric 2007jbv  Atanasiu Gh., Brinzei N. // "Transilvania" University, Brasov, Romania, gh_atanasiu@yahoo.com, nico.brinzei@rdslink.ro
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 BerwaldMoor metric, and to metrics conformally related to it.
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On the World function and the relation between geometries 2006jaz  Garas`ko G. I. // Electrotechnical Institute of Russia, Moscow,
gri9z@mail.ru}
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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2005jbx  Pavlov D. G.
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2005jbw
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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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The notions of distance and velocity modulus in the linear Finsler spaces 2005jaz  Garas'ko G. I., Pavlov D. G.
The formulas for the 3dimensional distance and the velocity modulus in the
4dimensional linear space with the BerwaldMoor metrics are obtained. The used algorithm is applicable both for the Minkowski space and for the arbitrary polylinear Finsler space in which the timelike component could be chosen. The constructed here modulus of the 3dimensional velocity in the space with the BerwaldMoor metrics coincides with the corresponding expression in the Galilean space at small (nonrelativistic) 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 3dimensional 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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2005jay
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The normal conjugation on the polynumber set. 2004jby  Garasko G.I., Pavlov D.G.
The polynumber set is an example of linear space with several polylinear forms. The concept of normal conjunction is introduced on the set of nondegenerated nnumbers. The normal conjunction is a (n1)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 nnumber 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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Generalizedanalytical functions and the congruence of geodetic. 2004jbx  Garas'ko G. I.
Some properties of generalizedanalytical functions of polynumber 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 generalanalytic 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 generalizedanalytical function itself.
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Generalization jf Scalar Product Axioms 2004jay  Pavlov D. G.
The concept of scalar product is vital in studying basic properties of
either Euclidean or pseudoEuclidean spaces. A generalizing of a special
subclass 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 fourdimensional linear Finslerian space related to the algebra of commutativeassociative hypercomplex numbers, that are called Quadranumerical, are proved in the concrete polyform.
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Chronometry of the threedimensional time 2004jax  Pavlov D. G.
The concept of the multidimensional 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 twodimensional case) example of three temporial dimensions.
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Fourdimensional time 2004jaw  Pavlov D. G.
The generalized metric space, that can be called the flat fourdimensional time, is based on the BerwaldMoore'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 threedimensional distance, the speed, etc. It is
demonstrated how from the point of the physical observer, associated with the
world line, in absolutely symmetrical fourdimensional 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 threedimensional space which at the limit is the Euclidean space.
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Properties of spaces associated with commutativeassociative H_{3} and H_{4} algebras 2004jau  Lebedev S. V.
In the first part of this work a real axis of the space associated with
the H_{3} 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 H_{4} algebra are presented.
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Generalizedanalytical functions of polynumber variable 2004jat  Garas`ko G. I.
We introduce the notion of the generalizedanalytical function of the polynumber variable, which is a nontrivial 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 associativecommutative hypercomplex numbers H_{4} and an interesting class of corresponding functions.
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Threenumbers, which cube of norm is nondegenerate threeform 2004jap  Garas`ko G. I.
Arbitrary threeform can be put in a canonical form. The requirement of
existence of twoparametric Abelian Lie group to play the role of group of
symmetry for threeform admits selecting the threeforms that correspond to threenumbers and finding all the threenumbers which cube of norm is a nondegenerate threeform with respect to a special coordinate system. There are exactly two (up to isomorphism) such sets of hypercomplex numbers, namely the sets: C_{3}, H_{3}. They can be regarded as generalizations of complex and binary (hyperbolic) binumbers to the case of threenumbers.
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