"Hypercomplex Numbers in geometry and Physics" 2 (14), Vol 7, 2010
j014

Content of Issue

About V International Educational Workshop «Fundamentals of Finsler Geometry and it's Applications in Physics»
2010jaz | Pavlov D.G.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, geom2004@mail.ru

Editor's note

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Algebraic unified theory of space-time and matter on the double variable plane
2010jbz | Pavlov D.G., Kokarev S.S.   // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia; RSEC Logos, Yaroslavl, Russia, geom2004@mail.ru, logos-distant@mail.ru

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

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

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

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Jet local Riemann-Finsler geometry for the three-dimensional time
2010jdz | Atanasiu Gheorghe, Neagu Mircea  // University Transilvania of Brasov, Brasov, Romania, gh_atanasiu@yahoo.com, mircea.neagu@unitbv.ro

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

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Equations of electromagnetism in some special anisotropic spaces. Part 2
2010jez | Voicu Nicoleta  // Transilvania University, Brasov, Romania, nico.brinzei@unitbv.ro

By using variational calculus and exterior derivative formalism, we proposed in [1] a new geometric approach to electromagnetism in spaces with metrics obtained as small deformations of flat Finsler metrics. The ideas were extended to general Finsler spaces in [11]. In the present paper, we provide more details regarding generalized currents, the domain of integration and gauge invariance. Also, for flat Finsler spaces, we define the generalized energy-momentum tensor as the symmetrized Noether current corresponding to the invariance of the field Lagrangian with respect to spacetime translations.

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Galilean nilpotent spaces of dimension 3 with 2-dimensional time. Geometric properties
2010jfz | Dolgarew A.I., Dolgarew I.A.  // Penza State University, Penza, Russia, delivar@yandex.ru

Studied the curves and surfaces. Determined curvature, torsion curves do not possess. Proved definability of the curve function of its curvature. Considered time and space-time surface, defined by their first and second quadratic forms, the total curvature. Proved definability surface coefficients of their quadratic forms

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Axially symmetric generalization of the Cauchy-Riemann system and modified Clifford analysis
2010jgz | Bryukhov D.A.  // Fryazino, Russia, bryukhov@mail.ru

The main goal of this paper is to describe the most adequate generalization of the Cauchy-Riemann system fixing properties of classical functions in the octonionic case. An octonionic generalization of the Laplace transform is introduced. Octonionic generalizations of the inversion transformation, of the Euler gamma function and of the Riemann zeta-function are given.

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To the theory of a physical vector field in natural three-dimensional space for geometry of events Berwald-Moor
2010jhz | Zaripov R.G.  // Institute of Mechanics and Engineering, Kazan Science Center, Russian Academy of Sciences, Kazan, Russia, zaripov@mail.knc.ru

The model of a physical vector field with density of scalar and vector sources in natural three-dimensional space for geometry of events Berwald-Moor are is considered. The density of an energy and its stream which depend on second derivative of components of a vector of strength are defined. Expression for the force working on a source of a field is deduced and the equations of motion of the charged particle are submitted. The question of waves of a field of "deformations"in vacuum is discussed.

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Bianchi type identities in generalized Finsler space
2010jiz | Zlatanovic Milan Lj., Mincic Svetislav M.  // University of Nis, Faculty of Science and Mathematics, Nis, Serbia, zlatmilan@yahoo.com, svetislavmincic@yahoo.com

In the work [4, 14] we have studied a generalized Finsler space (GFN) (with non-symmetric basic tensor) and have obtained four curvature tensors by using four kinds derivatives in the sense of Runds ƒ-differentiation. In the present work we study Bianchi type identities, related to mentioned curvature tensors in GFN, generalizing the known Bianchi identity from the usual Finsler space.

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Multidimensional laplace transforms over Cayley-Dickson algebras and their applications
2010jjz | Ludkovsky S.V.   // Moscow State Technical University MIREA, Moscow, Russia, sludkowski@mail.ru

Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.

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Benoit Mandelbrot: the way to fractal geometry of nature
2010jkz | Panchelyuga V.A.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia; Institute of Theoretical and Experimental Biophysics of the RAS, Pushchino, Russia, panvic333@yahoo.com

October 14, 2010 in Cambridge, Massachusetts died founder of fractal geometry Benoit Mandelbrot at the age of 85. Following paper is writing in memory of prominent scientist.

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