"Hypercomplex Numbers in geometry and Physics" 1 (15), Vol 8, 2011
j015

Content of Issue

GEOMETRICAL MODELS OF THE LOCALLY ANISOTROPIC SPACE-TIME
2011jaw | Balan V., Bogoslovsky G.Yu., Kokarev S.S., Pavlov D.G., Siparov S.V., Voicu N.  // University Politehnica of Bucharest, Bucharest, Romania; Skobeltsyn Institute of Nuclear Physics, MSU, Moscow, Russia; Research Institute of Hypercomplex Systems in Geometry and Physics, Fryazino, Russia; RSEC Logos, Yaroslavl, Russia; Research Institute of Hypercomplex Systems in Geometry and Physics State University of Civil Aviation, St-Petersburg, Russia; "Transilvania" University of Brasov, Brasov, Romania, vladimir.balan@upb.ro, bogoslov@theory.sinp.msu.ru, logos-center@mail.ru, geom2004@mail.ru, sergey@siparov.ru, nico.brinzei@unitbv.ro

Lately, the problem of the Lorentz symmetry-breaking has been widely discussed in literature. It is worthy to note that, in addition to the construction of phenomenologically focused effective field theories, the research based on the Finslerian geometric models of space-time becomes more and more popular. Finsler approach to the Lorentz symmetrybreaking problem is characterized by the fact that there the Lorentz symmetry-breaking is not accompanied by the relativistic symmetry-breaking. This means that the preservation of the relativistic symmetry is a rigid criterion of the viability for any non-Lorentzinvariant effective field theory. Though this paper has a review character, it mainly contains original results obtained by the authors, concerning Finsler extensions of Relativity Theory.

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INDUCED REPRESENTATIONS AND HYPERCOMPLEX NUMBERS
2011jbw | Kisil V.V.  // University of Leeds, Leeds, Great Britain, kisilv@maths.leeds.ac.uk

We review the construction of induced representations of the group G = SL2(R). Firstly we note that G-action on the homogeneous space G/H, where H is any one-dimensional subgroup of SL2(R), is a linear-fractional transformation on hypercomplex numbers. This observation can be extended to further correspondences between structural components of SL2(R) and hypercomplex systems. Thus we investigate various hypercomplex characters of subgroups H. In particular we give examples of induced representations of SL2(R) on spaces of hypercomplex valued functions, which are unitary in some sense.

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EXTRA-VARIATIONAL PRINCIPLE IN THE THEORY OF FIELD
2011jcw | Kokarev S.S.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia; RSEC Logos, Yaroslavl, Russia, logos-center@mail.ru

We formulate natural algorithm for calculation of fundamental constants and Largrangians within every fundamental theory, which can be derived from standard variational principle. We illustrate the method by multiple examples in classical mechanics, generalized gravitational theories, unified h-holomorpfic theories, law dimensional and high dimensional field-theoretical models.

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SEVERAL REMARKS ON THE ANISOTROPIC GEOMETRODYNAMICS
2011jdw | Siparov S.V.  // State University of Civil Aviation, St-Petersburg, Russia; Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, sergey@siparov.ru

The paper contains the discussion of several important aspects of the approach suggested earlier (anisotropic geometrodynamics), the analysis of the disagreement between the GRT predictions and some observations on the galactic scale, and the alternative interpretation of the observations related to the hypothesis of the dark matter existence in the gravitational lenses.

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POLYADIC OPERATIONS ON THE CARTESIAN POWERS OF GROUPOIDS, SEMIGROUPS AND RINGS
2011jfw | Galmak A.M.  // Mogilev State University of Food Technology, Mogilev, Belarus, mgup@mogilev.by

Previously was shown [2] that for any numbers n ≥ 3, s ≥ 1, m ≥ 2, on the Cartesian powers An−1 and Am(n−1) of the semigroup A, are respectively defined the (s(n−1)+1)−ary operation [ ]s(n−1)+1,n−1, and the n−ary operation [ ]n,m,m(n−1). Also was shown that for any three integers k ≥ 2, l ≥ 2 and m ≥ 1, and any permutation ƒ 2 Sk on the Cartesian power Bmk of the set B, is defined an l−nary relation [ ]l,ƒ,m,mk. In the present paper studies concerning the polyadic operations on the cartesian powers of universal algebras are continued.

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THE SPACE OVER THE ALGEBRA OF POLYNUMBERS WITH BERWALD-MOOR METRIC
2011jiw | Bukusheva A.V.  // Saratov State University, Saratov, Russia, bukusheva@list.ru

On a smooth manifold is defined a poly-affinor algebra compatible with a Berwald-Moor metric. We define the conditions under which a manifold with a poly-affine algebra can be equipped with the structure of a space over the algebra of polynumbers.

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CONSTRUCTION OF THE CURVATURE ON CAYLEY ALGEBRA OCTAVES
2011jgw | Dorofeev V.Yu.  // Friedmann Laboratory for Theoretical Physics, St-Petersburg, Russia, friedlab@mail.ru

Lagrangian of matter fields of electron-lepton sector is constructed on Cayley algebra octaves. It is inferred from the model that the state space is ten-dimensional. It is proposed to determine the non-associative part of the Lagrangian as a manifestation of gravity. It turns out a pair of oppositely charged massless vector bosons induces gravity. It is shown the model of the Lagrangian with Schwarzschild as well as Friedman metric is consistent.

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FINSLER GEOMETRY EFFECTS IN SURFACE PHENOMENA PHYSICS: THE CASE OF MONOLAYER SYSTEMS
2011jkw | Grushevskaya H.V., Krylova N.G.  // Belarusan State University, Minsk, Belarus, grushevskaja@bsu.by, nina-kr@tut.by

Interactions in a LB-monolayer have been geometrized in such a way that solutions of equations for a motion of particles in the monolayer are approximated by geodesics of Finsler two-dimensional space. Two-dimensional Finsler metric effects in surface phenomena physics for the monolayer case were investigated. The calculations of model proposed to geometrize interactions at LB-monolayer formation were carried out in a resonance approximation. A simulation has shown that there exist several regimes of the structure formation which depend on compression speed and characteristics of double electrical layer.

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ANISOTROPY OF RED SHIFT
2011jlw | Levin S.F.  // Moscow Institute for expertise and tests, Moscow, Russia, info@rostest.ru, antoninaEL@rostest.ru

Statistical and metrological aspects of measurement date analysis problem for research of anisotropy of radiation of extra galaxies source have been considered. As illustrated in the review, the reason for dipoles anisotropy of cosmic background radiation and red shift of galaxies, radio galaxies and quasars is large-scale of universe heterogeneity.

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ON POSSIBLE DEPARTURE DETECTED VELOCITY OF PROPAGATION OF FUNDAMENTAL INTERACTIONS FROM LIGHT VELOCITY IN VACUUM
2011jmw | Gladyshev V.O.  // Bauman Moscow State Technical University, Moscow, Russia; Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, vgladyshev@mail.ru

It was shown that for the fundamental interactions, which propagate in a space of independent variables, results of measuring procedure, based on measurements of partial differentials of transformations, depend on laboratory inertial frame velocity relative to a space of propagation of interactions. Possible interpretation for invariance violation on the detector OPERA and in the experiment on measuring the time of a neutrino splash from SN1987A with neutrino and gravitational detectors was given. The estimation method of space anisotropy was offered. It is based on measurement of signal variations when the laboratory set changes its orientation in a space. It was pointed out that the main effective method of determining the anisotropy parameter is the experiment SADE suggested in the works [1, 2]. In the experiment space anisotropy was estimated on results of interference detection for velocity of electromagnetic radiation propagation in a rotating optical disk.

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