The articles on hyper-complex numbers, Finslerian geometry and physics.
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The articles given bellow meet one of the following demands: – have been printed in “Hyper-complex number in geometry and physics”
– have been sent for publication on this site by their authors
– have been translated to be published in the journal or on this site.
– Appear to be important enough for this site
The least materials on hyper-complex numbers, Finslerian geometry and physics are placed in the “Archive” section.

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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The equations of electromagnetism in some special anisotropic spaces
2008jbw | Nicoleta Brinzei & Sergey Siparov  // Transilvania University, Brasov, Romania \\ Academy of Civil Aviation, St. Petersburg, Russia; nico.brinzei@rdslink.ro, sergey@siparov.ru

We show that anisotropy of the space naturally leads to new terms in the expression of Lorentz force, as well as in the expressions of currents.

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On the possibility of the OMPR effect in spaces with Finsler geometry. Part II
2008jbv | Nicoleta Brinzei & Sergey Siparov   // Transilvania University, Brasov, Romania \\ Academy of Civil Aviation, St. Petersburg, Russia; nico.brinzei@rdslink.ro, sergey@siparov.ru

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.

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Spectral properties and applications of the numerical multilinear algebra of m-root 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 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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Modern higher energy physics
2008jav | K. V. Stepanyantz  // Russia, Moscow State University, physical faculty, department of theoretical physic, sstepan@phys.msu.ru

We briefly review the modern higher energy physics. Description of the strong and electroweak interactions is considered both in the Standard model and in Grand Unification theories. We discuss the origin of the small neutrino mass and the necessity of supersymmetry. We also consider description of the gravity, modern cosmological experimental data, supergravity theories, and a relation of gravity with other interactions. We draw attention to problems, which appear describing the gravity, and possible ways of their solution. In particular, achievements and drawbacks of superstrings theories are discussed.

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Poly-Numbers (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 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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On the possibility of the OMPR effect in the space with Finsler geometry. Part 1.
2007jbw | Brinzei N., Siparov S. V.  // "Transilvania" University, Brasov, Romania, nico.brinzei@rdslink.ro, & State University of Civil Aviation, Sankt-Petersburg, Russia, sergey@siparov.ru

The effect of the optic-metrical parametric resonance could provide the possibility to obtain the experimental evidence of the gravitational waves existence. The effect might change, if the geometry of the physical space-time is not Riemannian but Finslerian one. The investigation of this situation is undertaken.

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Einstein Equations for the Homogeneous Finsler Prolongation to TM, with Berwald-Moor 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 Berwald-Moor metric, and to metrics conformally related to it.

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An extension of electrodynamics theory to complex Lagrange geometry
2007jbu | Gh. Munteanu  // Transilvania Univ., Faculty of Mathematics and Informatics, Bra\c{s}ov, Romania, gh.munteanu@unitbv.ro

In this note our purpose is to introduce the Maxwell type equations in a complex Lagrange space, particularly in a complex Finsler space. The electromagnetic tensor fields are defined as the sum between the differential of the complex Liouville 1-form and the symplectic 2-form of the space relative to the adapted frame of Chern-Lagrange complex nonlinear connection. Is proved that the (1,1)-type electromagnetic field of a complex Finsler space vanish and the differential of the (2,0)-type electromagnetic field yields the generalized Maxwell equations. The complex electromagnetic currents are also introduced and the conditions when they are conservative are deduced. Finally we apply the results to the electrodynamics Lagrangian considered in [Mu] and to the case of complex Randers spaces.

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Stumbling blocks for standard cosmology in the light of six-dimensional one
2007jbp | I. A. Urusovskii  // Acoustic institute n.a, acad. N.N. Andreev, Moscow, Russia

An account of an increase of speed of light in the actual three-dimensional Universe and its effect on redshift for distant sources and on theoretical redshift dependencies compared with observed data is given. The investigation is carried out on the basis of the simplest six-dimensional treatment of the expanding Universe in the form a three-dimensional sphere appeared as a result of the intersection of three simplest geometrical objects of finite dimensions in the six-dimensional Euclidean space -- of three uniformly expanding five-dimensional spheres. A scenario in which the speed of light (and the energy of each elementary particle) in the six-dimensional space is constant in time is considered. Some difficulties of standard cosmology are discussed on the base of six-dimensional cosmology.

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Inertial navigation equations and the quaternion space-time theory
2007jaq | V. F. Chub  // Korolev Rocket-Space Corporation ``Energiya'', Russia

The comparative analysis of relativistic and nonrelativistic inertial navigation equations in space free of gravitational field was performed in review. In order to represent the equations, the use is made of the quaternions with real, dual, complex and complex-dual factors. Within the theory based upon the quaternions with complex-dual factors (the quaternion space-time theory) it was demonstrated that the transformations forming in the context of a special relativity theory Poincare group and in the context of classical Newton mechanics Galilei group were open. The inertial navigation equation consistent with the quaternion space-time theory was given and its absurdity was noted.

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A mathematical description of the fermionic state
2007jap | Rowlands P.  // Department of Physics, University of Liverpool, Oliver Lodge Laboratory, Liverpool, UK, p.rowlands@liverpool.ac.uk

The fermionic state is the foundation for the whole of physics. Physics is entirely concerned with fermions and their interactions, and nothing else. It is possible to derive a mathematical expression for the fermionic state, which is an operator only, not an equation, or wavefunction. This operator appears to contain within it all the information needed to construct fermion interactions and particle states. Extensions to particle representations using Finsler geometry could find this formalism a particularly accessible link.

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Finsler spaces with polynomial metric
2006jbt | L. Tamassy

In this paper we want to show that Finsler spaces with polynomial metric allow metrical tensorial connections (linear for a given type of tensors). Many of them induce, in a natural way, metrical non-linear connections in $\tau_M$.

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Pairs of metrical Finsler structures and Finsler connections compatible to them
2006jbs | Atanasiu Gh.

We consider a pair of metrical Finsler structure $g_{ij}\left( x,y\right),s_{ij}\left( x,y\right) , \left( x,y\right) \in TM,\;i,j=\overline{1,n},\dim M=n$ and we investigate the cases in which is possible to find Finsler connections compatible to them:$\;rank\left\| g_{ij}\left( x,y\right) \right\| =n,$ $rank\left\| s_{ij}\left( x,y\right) \right\| =n-k,\;k\in\left\{ 0,1,...,n-1\right\} ,\forall \left( x,y\right) \in TM\setminus \left\{ 0\right\} .$

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The horizontal and vertical semisymmetric metrical $d$-connections in the Relativity Theory
2006jbr | Atanasiu Gh., Stoica E.

Let $E$ be the $(m+n)$-dimensional total space of a vector bundle $(E,p,M)$, $dim\;M=n$, a given fixed nonlinear connection $N$ on $E$ and a given $(h,v)$-metrical structure $G\in \mathcal{T}_{2}^{0}\left( E\right) $. In the paper, we determine the Einstein equations of an $h$- and $v$-semisymmetric metrical distinguished connection on $E=TM$, if $n=4$, for a Riemann -- local Minkowski model.

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CMC and minimal surfaces in Berwald-Moor spaces
2006jbq | Balan V.

For Randers and Kropina Finsler spaces are described the extended equations of minimal and CMC hypersurfaces. For the Berwald-Moor type Finsler metric are then considered different types of symmetric polynomials generating the fundamental function and classes of CMC surfaces are evidentiated. Maple 9.5 representations of indicatrices point out structural differences among Berwald-Moor fundamental functions of different order, leading to different CMC approaches.

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Geodesics, connections and Jacobi fields for Berwald-Moor quartic metrics
2006jbp | Balan V., Brinzei N., Lebedev S.

For Finsler spaces $(M,F)$ with quartic metrics $F=\sqrt[4]{G_{ijkl}(x,y) y^{i}y^{j}y^{k}y^{l}},$ we determine the equations of geodesics and the corresponding arising geometrical objects-canonical spray, nonlinear Cartan connection, Berwald linear connection -- in terms of the non-homogenized flag Lagrange metric $h_{ij}=G_{ij00}.$ Further, are studied the geodesics and Jacobi fields of the tangent space $TM$ for $hv$-metric models.

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The Lagrangian-Hamiltonian formalism in gauge complex field theories
2006jbo | Munteanu Gh.

An introduction in the study of gauge field theory in terms of complex Finsler geometry on the total space of a $G$-complex vector bundle $E$ was made by us in \cite{Mu2}. Here we briefly recal the obtained results and similar notions are investigated on the dual bundle $E^{*}$ by complex Legendre transformation (the $\mathcal{L}$-dual process).
The complex field equations are determined with respect to a gauge complex vertical connections. The complex Hamilton equations are write for the general $\mathcal{L}$-dual Hamiltonian obtained as a sum of particle Hamiltonian, Yang-Mills and Hilbert-Einstein Hamiltonians.

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Some geometrical aspects of harmonic curves in a complex Finsler space
2006jbn | Munteanu Gh.

In this note we make a short study of the geometry of curves in a complex Finsler space. For harmonic curves we obtain an equivalent characterization to that from \cite{Ni}. A special discussion concerns the holomorphic curves.

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Fundamental equations for a second order generalized Lagrange space endowed with a Berwald-Moor type metric in invariant frames
2006jbm | Paun M.

The purpose of this paper is to study Vranceanu identities and Maxwell equations of a generalized Lagrange space of order 2 endowed with a Berwald Moor type metric in invariant frames end to emphasize their equivalence.

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Theory of the zero-order effect to investigate the space-time geometrical structure
2006jbl | S. V. Siparov

The applicability of Einstein's Relativity Theory on the galactic scale and the role of geometry in the problems of astrophysical observations are discussed. The theory of the zero-order effect to study experimentally the geometrical properties of space-time is suggested

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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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Construction of the pseudo Riemannian geometry on the base of the Berwald-Moor geometry
2006jay | Garas`ko G. I., Pavlov D. G.  // gri9z@mail.ru; geom2004@mail.ru

The space of the associative commutative hyper complex numbers, H₄, is a 4-dimensional metric Finsler space with the Berwald-Moor metric. It provides the possibility to construct the tensor fields on the base of the analytical functions of the H₄ variable and also in case when this analyticity is broken. Here we suggest a way to construct the metric tensor of a 4-dimensional pseudo Riemannian space (space-time) using as a base the 4-contravariant tensor of the tangent indicatrix equation of the Berwald-Moor space and the World function. The Berwald-Moor space appears to be closely related to the Minkowski space. The break of the analyticity of the World function leads to the non-trivial curving of the 4-dimensional space-time and, particularly, to the Newtonian potential in the non-relativistic limit

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2005jbx | Pavlov D. G.

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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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Finsler spinors
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The prolongations of a Finsler metric to the tangent bunde $T^k(M) (k>1)$ of the higher order accelerations
2005jbq | Atanasiu Gh.  // Department of Algebra and Geometry, Transilvania University, Brasov, Romania

An old problem in differential geometry is that of prolongation of a Riemannian structure $g\left( x\right) $ on a real $n-$dimensional $% C^{\infty }$-manifold $M,x\in M,$ to the bundle of $k-$jets $\left( J_{0}^{k}M,\pi ^{k},M\right) $ or, equivalently the tangent bundle $\left( T^{k}M,\pi ^{k},M\right) $ of the higher order accelerations. The problem belongs to so-called geometry of higher order. It was solved in $\left[ 18% \right] $ for $k=1$ and partially in $\left[ 19\right] $ for$\;k=2.$ The same problem of prolongation can be considered for a Finslerian structure $% F\left( x,y^{\left( 1\right) }\right) $. In the paper $\left[ 15\right] $ are given these solutions in the general cases, using the Sasaki-Matsumoto $N-$lift (for $k=2,$ see $\left[ 3\right] $ and $\left[ 6\right] ).

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The 2-Cotangent Bundle with Berwald-Moor Metric
2005jbp | Gheorghe Atanasiu, Vladimir Balan  // Transilvania University, Brasov, Romania; University Politehnica of Bucharest, Department Mathematics I, Romania

On the total space of the dual bundle $(T^{\ast 2}M,\pi ^{\ast 2},M)$ of the $2-$tangent bundle $(T^{2}M, \pi ^{2},M)$, the paper develops results related to the notions: of nonlinear connection, distinguished tensor fields, almost contact structure, Riemannian structures, $N-$linear connections and associated convariant derivations. The Ricci identities are derived and the local expressions of the corresponding $d-$tensors of torsion and curvature are provided. Further, the metric structures and the metric $N-$linear connections are studied, and the obtained results are specialized to the case when the metric tensor field is of Berwald-Moor type.

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The Berwald-Moor metric in the tangent bundle of the second order
2005jbo | Gheorghe Atanasiu, Nicoleta Brinzei

As an application of the results of the first author obtained in the papers \cite{1} and \cite{2}, the geometry of the second order tangent bundle $% T^{2}M$ (or second order jet bundle $J_{0}^{2}M$) endowed with two special types of metrics compatible with the 2-contact structures is studied. The particularity of these two models is that the horizontal and the $v^{(1)}$-\ part of the metric are both given by the same Riemannian metric (respectively, its horizontal part is Riemannian), while its $v^{(2)}$-part is given by the flag-Finsler Berwald-Moor metric (respectively, the $v^{(1)} $ and $v^{(2)}$- parts are given by the flag-Finsler Berwald-Moor metric, \cite{Mangalia}).

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Berwald-Moor-type $(h,v)$-metric physical models
2005jbn | Balan V., Brinzei N.  // University Politehnica of Bucharest, Department Mathematics I; Department of Mathematics, "Transilvania" University, Brasov, Romania

In the framework of vector bundles endowed with $(h,v)-$metrics several physical models for relativity are presented. A characteristic of these models is that the vertical part is provided by the flag-Finsler Berwald-Moor (fFBM) metric, while the horizontal part is specialized to the conformal and to Synge-relativistic optics metrics. As well, the particular case of $h-$Riemannian $v-$fFBM metric of Riemann-Minkowski type is examined, considering as nonlinear connection both the trivial canonical connection, and the one induced by the Lagrangian of electrodynamics. For all these models, basic properties are described and the extended Einstein and Maxwell equations are determined.

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Invariant frames for a generalized Lagrange space with Berwald-Moor metric
2005jbm | Marius Paun  // Faculty of Mathematics and Informatics, Transilvania, University of Brasov, Romania

The notion of generalized Lagrange space should be geometrically considered as a generalized metric space $M^n=(M,g_{ij}(x,y))$. A theory of invariant Finsler spaces was given by M. Matsumoto and R. Miron with important applications. The notion of non-holonomic space was introduced by Gh. Vranceanu in [VR]. The Vranceanu type invariant frames and the invariant geometry of second order Lagrange spaces was studied by the author in [P3]. The purpose of the present paper is to study the invariant geometry for a generalized Lagrange space endowed with a Berwald-Moor metric. We introduce distinct non-holonomic frames on the two components of the Whitney's decomposition. This will determine a non-holonomic coordinates system on the total space $TM$ and thus its geometry can be studied with methods analogous to the mobile frame. We obtain, in this manner, invariant connections, curvatures and torsions, and the fundamental equations in this theory. Also we can construct the invariant frames so that, with respect to them, the metric of the total space can be written in canonical form and in this case we deduce invariant Einstein equations. We mention that the frames introduced here depend on the metric and all the computations are for this metric.

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2005jbl

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Expansion of Complex Number
2005jbk | Y. A. Furman, A. V. Krevetsky  // Mari state technical university, Yoshkar Ola, Russia

The expanded complex numbers are introduced by means of imaginary unit $i$ replacement by one-dimensional on multivariate $3D$ or $7D$ imaginary unit $r$. It is shown, full quaternions and octaves appear as a result of a turn around the material axis $0Re$ plane where $a+ib$ number is set in $4D$ and $8D$ spaces. Rotor-complanar classes of quaternions and the octaves appearing as a result of similar transformations are considered. They represent commutative-associative algebras.

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Thomas precession by pseudoquaternions
2005jbj | D. E. Burlankov, G. B. Malykin  // Nizhny Novgorod State University; Institute of Applied Physics RAS, Nizhny Novgorod, Russia

When a body moves curvilinearly in a plane with a velocity that is comparable to the velocity of light, only three coordinates of the body undergo Lorentz transformation, and the transformation matrix appears to be three-parametric. This enables description of these transformations by pseudoquaternions, Hamilton quaternions slightly modified for pseudo-Euclidean character of the metrics. Their algebraic properties and relation to the Lorentz transformations in a 2+1-dimensional Minkovsky space were determined. We integrated the pseudoquaternion differential equation of continuous transformations at a body's motion along a circular orbit and, as a result, obtained an expression for the value of the Thomas precession.

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Nonclosure of elemenary space-time transformations
2005jbi | Chub V. F.  // Korolev Rocket and Space Corporation "Energia"

The article gives a brief group-theoretical comparative study of three space-time theories: (space-time theory in frames of) classical Newton mechanics, special theory of relativity and author-developed theory based on complex-dual quaternions.

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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 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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2005jay

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2005jax

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2005jaw

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2005jav

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2005jau

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2005jat

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2005jas

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Conference “Number, time, relativity”
2004jbz | Gladyshev V.O., Pavlov D.G.

On the 13th august of 2004 the International scientific conference “Number, time, relativity” took place in N. E. Bauman Moscow State Technical University. The purposes of the conference were: to attract the attention of foreign and Russian physicists to Finslerian generalizations of the relativistic theory, to gather the leading specialists in the field of hyper-complex numbers, Finslerian geometry (that generalize the Riemannian manifolds), and the specialists in the field of the relativistic theory. The conference was devoted to 175th anniversary of N. E. Bauman Moscow State Technical University. The conference was performed by: the Bauman University’s cathedra of physics, the theoretical physics cathedra of Moscow State University of M.V. Lomonosov and the United Physics Society of Russian Federation. The main sponsor of the conference was the Fund of 175th anniversary of N. E. Bauman Moscow State Technical University.

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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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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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Concerning the norm of biquaternions and some other algebras with central conjunction
2004jbw | Eliovich A. A.

The concept of central conjunction is introduced in this article. We apply it to algebras of biquaternions and bioctaves. With the given analysis method of the conjunctions permitted by algebra we derive some new results. Thus the alternative algebras with central conjunction are proven to have the multiplicative norm of second degree (that is in general not real). The consequence of this fact is that these algebras (biquaternions and bioctaves particulary) have the multiplicative real norm of degree higher than 2. This norm has several different but equivalent views. The quadrascalar and quadravector multiplications are introduced. Some results for algebras of biquternions, diquaternions and bioctaves are given in terms of isotropic basis. The developed methods may be useful in the geometrical and physical usage of concerned algebras.

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On some distributive algebras
2004jbv | Solovey L.G.

The examined type of sets, that are not rings, but in some sense are close to them. These sets are called 'Hyper-rings'. They consist of several additive groups, that intersect each over at the zero only. Yet, they are multiplicative groupoids (or groups, excluding the zero). The distributive laws are fulfilled.
Rings (and in particular the bodies and the fields) are the special case of the concerned sets. The given examples witness that such sets are highly disseminated. So, the idea that the real physical values may be "layed" in the ring is wrong, because they are subset of hyper-ring.
The real hyper-rings with unity can not be reduced to rings. Their additive groups are vector spaces, and they may be treated as a generalized Hyper-complex systems, in which we include the real binary (provided with summation and multiplication) distributive algebraic structures with neutral element, where the number of included vector spaces is more than one and finite.
The example of hyper-rings, suggesting that scrutiny of them is worth-able, are the second order matrices, that are mostly like unitary matrixes, but normalized not by unity. They are normalized by an arbitrary non-negative number. The complex numbers and the quaternions may be represented with such matrixes while they are the ones subspace.

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Deformation principle as foundation of physical geometry and its application to space-time geometry
2004jbu | Rylov Yuri A.

Physical geometry studies mutual disposition of geometrical objects and points in space, or space-time, which is described by the distance function $% d$, or by the world function $\sigma =d^{2}/2$. One suggests a new general method of the physical geometry construction. The proper Euclidean geometry is described in terms of its world function $\sigma _{\mathrm{E}}$. Any physical geometry $\mathcal{G}$ is obtained from the Euclidean geometry as a result of replacement of the Euclidean world function $\sigma _{\mathrm{E}}$ by the world function $\sigma $ of $\mathcal{G}$. This method is very simple and effective. It introduces a new geometric property: nondegeneracy of geometry. Using this method, one can construct deterministic space-time geometries with primordially stochastic motion of free particles and geometrized particle mass. Such a space-time geometry defined properly (with quantum constant as an attribute of geometry) allows one to explain quantum effects as a result of the statistical description of the stochastic particle motion (without a use of quantum principles).

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The Nilpotent Vacuum
2004jbt | Rowlands Peter

A fermionic state vector which is a nilpotent or square root of zero appears to be the most convenient packaging of the fundamental physical parameters space, time, mass and charge into a single unit. It also has the advantage of being a supersymmetric quantum field operator, which uniquely and simultaneously specifies both amplitude and phase for any fermionic state, and incorporates all the specific aspects required in BRST field quantization into a single package. The mathematical structure of the state vector immediately generates vacuum terms relevant to all four fundamental interactions, and explains the symmetry-breaking between them. By incorporating the vacuum aspects into our understanding of the fermion, we generate a ‘string theory without strings’. The nilpotent vacuum operators suggest links with many well-known vacuum phenomena, including the Casimir effect and zero-point energy.

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Division Algebra, Generalized Supersymmetries and Octonionic M-Theory
2004jbs | Toppan Francesco

This is the report of the talk given at the conference ``Number, Time and Relativity", held at the Bauman University, Moscow, August 2004, concerning the recent research activity of the author and his collaborators about the inter-relation of the concepts of division algebras, representations of Clifford algebras, generalized supersymmetries with the introduction of an alternative description of the M-algebra in terms of the non-associative structure of the octonions.

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From Editorialboard
2004jaz | Editorialboard of Journal

Number is one of the most fundamental concepts not only in mathematics, but in general natural science as well. It may be primary even in comparison with such global categories as time, space, substance, matter, and field. That is why editing the first issue of the journal "Hypercomlex numbers in geometry and physics" the editorial board sincerely hopes that articles not only on numbers in general, but primarily the works that reveal their organic connection with the real world will find here their true scope.

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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 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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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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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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Finsleroid -- space supplemented by angle and scalar product
2004jav | Asanov G. S.

The science of the past century has achieved great success on the basis of the geometrical quadratic concepts that were followed as logical and mathematical primaries. More profound ideas will imply using a more capacious class of geometries, for example the Finsler one which inscribes structures because the Finslerian indicatrices are no more isotropic in all directions. In the present work an attempt is made to resolve the respective difficulties of Finsler generalization by choosing the particular Finsleroid--type metric that implies one preferred direction, admitting the total axial symmetry around it. In this case, interesting constructive methods of introducing the concept of the angle and scalar product outside the frame of the Euclidean Geometry can conveniently be opened up.

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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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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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Algebrodinamics: Primodial Light, Particles-Caustics and Flow of Time
2004jas | Kassandrov V. V.

In the field theories with twistor structure particles can be identified with (spacially bounded) caustics of null geodesic congruences defined by the twistor field. As a realization, we consider the ``algebrodynamical'' approach based on the field equations which originate from noncommutative analysis (over the algebra of biquaternions) and lead to the complex eikonal field and to the set of gauge fields associated with solutions of the eikonal equation. Particle-like formations represented by singularities of these fields possess ``elementary'' electric charge and other realistic ``quantum numbers'' and manifest self-consistent time evolution including transmutations. Related concepts of generating ``World Function'' and of multivalued physical fields are discussed. The picture of Lorentz invariant light-formed aether and of matter born from light arises then quite naturally. The notion of the Time Flow identified with the flow of primodial light (``pre-Light'') is introduced in the context.

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On some questions of four dimensional topology: a survey of modern research
2004jar | Mikhailov R. V.

Our physical intuition distinguishes four dimensions in a natural correspondence with material reality. Four dimensionality plays special role in almost all modern physical theories. High dimensional quantum fields theory and string theory are considered often together with their compactifications, i. e. the main space, describing the reality is a product of a four-dimensional manifold with some compact high-dimensional space. In this way we come to the well-known Kaluza-Klein model and ten-dimension superstring theory.
It is an interesting fact that the dimension four is a more complicated dimension from pure mathematical point of view. It seems that there is a contradiction with our intuition in understanding of the dimension concept, really, new dimensions give us new complexity. But it is not true in general. Additional dimensions often give a new freedom. It is natural that we must have some golden mean in this approach, in which we don't have a necessary freedom, but low-dimensional methods weakly work. In topology this mean is dimension four.
The goal of this note is to give a small survey of some problems in four-dimensional topology.

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Quaternions: Algebra, Geometry and Physical Theories
2004jaq | Yefremov A. P.

A review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups of Q-units transformations leaving Q-multiplication rule form-invariant are determined. A series of mathematical and physical applications is offered, among them use of Q-triads as a moveable frame, analysis of Q-spaces families, Q-formulation of Newtonian mechanics in arbitrary rotating frames, and realization of a Q-Relativity model comprising all effects of Special Relativity and admitting description of kinematics of non-inertial motion. A list of "Quaternionic Coincidences" is presented revealing surprising interconnection between basic relations of some physical theories and Q-numbers mathematics.

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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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On Possibility of Theoretical Disproof of Ether
2004eli | A. A. Eliovich  // PFUR

We discuss the problem: is it possible to disprove the ether conception on basis of classical electrodynamics equations, without appealing to experiment? We criticize recent T. A. Perevozskij's attempt to do it. We show that such mental experiments (with static fields and without changing of references frames) can't disprove even rough etherdynamical theories (although they can uncover unusual effects in such theories) and give no information for classical ether theories. After this we adduce the general arguments which show why it isn't possible to disprove ether on pure theoretical basis. We discuss the methodological context of the ether question and the notion of ether as interesting methodological instrument of natural science.

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Concerning the generalized Lorentz symmetry and the generalization of the Dirac equation
2004bg | G. Yu. Bogoslovsky, H. F. Goenner

The work is devoted to the generalization of the Dirac equation for a flat locally anisotropic, i.e., Finslerian space–time. At first we reproduce the corresponding metric and a group of the generalized Lorentz transformations, which has the meaning of the relativistic symmetry group of such event space. Next, proceeding from the requirement of the generalized Lorentz invariance we find a generalized Dirac equation in its explicit form. An exact solution of the nonlinear generalized Dirac equation is also presented.

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Concerning quaternions: finite displacements of solid bodies and points.
2002han | Khanukaev Yu. I.

We examine the quaternions technique as an alternative to vector and matrix description of spatial finite displacements of solid bodie. We also give the quaternionic description of Lorentz transformation.

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The Octonions
2002bae | Baez John C.  // Department of Mathematics University of California, baez@math.ucr.edu

From ArXiv:math.RA/0105155 v4 23 Apr 2002
The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.

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Finslerian Spaces Possessing Local Relativistic Symmetry
1999bgg | G. Yu. Bogoslovsky, H. F. Goenner

It is shown that the problem of a possible violation of the Lorentz transformations at Lorentz factors \gamma 5 x 10^10 , indicated by the situation which has dev eloped in the physics of ultra-high energy cosmic rays (the absence of the gzk cutoOE), has a nontrivial solution. Its essence consists in the discovery of the so-called generalized Lorentz transformations which seem to correctly link the inertial reference frames at any values of c. Like the usual Lorentz transformations, the generalized ones are linear, possess group properties and lead to the Einstein law of addition of 3-velocities. However, their geometric meaning turns out to be differen t: they serv e as relativistic symmetry transformations of a flat anisotropic Finslerian event space rather than of Minkowski space. Consideration is given to two typ es of Finsler spaces which generalize locally isotropic Riemannian space-time of relativit y theory, e.g. Finsler spaces with a partially and entirely broken local 3D isotropy. The investigation advances argumen ts for the corresp onding generalization of the theory of fundamen tal interactions and for a sp eci® c searc h for physical effects due to local anisotropy of space-time.

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On the possibility of phase transitions in the geometric structure of space-time
1998bbg | G. Yu. Bogoslovsky, H. F. Goenner

It is shown that space-time may be not only in a state which is described by Riemann geometry but also in states which are described by Finsler geometry. Transitions between various metric states of space-time have the meaning of phase transitions in its geometric structure. These transitions together with the evolution of each of the possible metric states make up the general picture of space-time manifold dynamics.

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1992ego

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Quaternionic analysis
1979sud | Sudbery Antony

The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real associative division algebra, namely the quaternions. Such a theory exists and is quite far-reaching, yet it seems to be little known. It was not developed until nearly a century after Hamilton's discovery of quaternions. Hamilton himself and his principal followers and expositors, Tait and Joly, only developed the theory of functions of a quaternion variable as far as it could be taken by the general methods of the theory of functions of several real variables (the basic ideas of which appeared in their modern form for the first time in Hamilton's work on quaternions). They did not delimit a special class of regular functions among quaternion-valued functions of a quaternion variable, analogous to the regular functions of a complex variable.
This may have been because neither of the two fundamental definitions of a regular function of a complex variable has interesting consequences when adapted to quaternions; one is too restrictive, the other not restrictive enough. The functions of a quaternion variable which have quaternionic derivatives, in the obvious sense, are just the constant and linear functions (and not all of them); the functions which can be represented by quaternionic power series are just those which can be represented by power series in four real variables.
In 1935 R Fueter proposed a deашnition of "regular" for quaternionic functions by means of an analogue of the Cauchy-Riemann equations. He showed that this definition led to close analogues of Cauchy's theorem Cauchy's integral formula, and the Laurent expansion. In the next twelve years Fueter and his collaborators developed the theory of quaternionic analysis.
The theory developed by Fueter and his school is incomplete in some ways, and many of their theorems are neither so general nor so rigorously proved as present-day standards of exposition in complex analysis would require. The purpose of this paper is to give a self-contained account of the main line of quaternionic analysis which remedies these deficiencies, as well as adding a certain number of new results. By using the exterior differential calculus we are able to give new and simple proofs of most of the main theorems and to clarify the relationship between quaternionic analysis and complex analysis.

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Forms permitting composition
1970sch | Schafer R. D.

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Forms degree n permiting composition
1963sch | Schafer R. D.

Important Article about hypercomplex numbers.

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