"Hypercomplex Numbers in geometry and Physics" 1 (17), Vol 9, 2012 j017
Content of Issue
POLYADIC OPERATIONS ON CARTESIAN POWERS OF CONJUGATE GROUP CLASSES 2012jaq | Galmak A.M., Vorobiev G.N., Balan V.D. // Mogilev State University of Food Technologies, Mogilev, Belarus; University Politehnica of Bucharest, Bucharest, Romania, mgup@mogilev.by, vbalan@mathem.pub.ro
For any l=3, k=2 and any permutation sigma from Sk on the Cartesian power A^k of a group A which admits a normal subgroup B, so that the factor group A/B be cyclic and having its order a divisor of l-1, we define the l-ary group , endowed with the l-ary operation [ ]l,sigma,k. We study the properties of this l-ary operation on Cartesian powers of conjugate group classes of the group A associated to its subgroup B.
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ABOUT POLYADIC OPERATIONS ON THE SPACEMATRICES SET 2012jbq | Galmak A.M. // Mogilev State University of Food Technology, Mogilev, Belarus, mgup@mogilev.by
For any integer number m=1 and any substitution on m symbols on all spacematrices sets, which have m section of any fixed orientation, defined partial polyadic operations. Also connection between this operations and polyadic operations on them m-component vector-matrices set is defined. Conditions of associability of defined polyadic operations are found. Transposed spacematrices are studied.
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ON THE FINSLERIAN MECHANICAL SYSTEMS 2012jcq | Miron Radu // University Transilvania of Brasov, Brasov, Romania
The notion of Finslerian Mechanical Systems was been introduced by author as a triple Sum F = (M, EF, Fe) formed by configuration space M, kinetic energy EF of a semidefinite Finsler space Fn=(M,F) and the external forces Fe. Fundamental equations of Sum F are the Lagrange equations. One determines the canonical semispray S and proves that the integral curves of S are the evolution curves of Sum F. Thus, the geometrical theory of the Finslerian mechanical systems Sum F can be studied by means of dynamical systems S on the velocity space TM.
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JET RIEMANN-HAMILTON GEOMETRIZATION FOR THE CONFORMAL DEFORMED BERWALD-MOOR QUARTIC METRIC DEPENDING ON MOMENTA 2012jdq | Oana Alexandru, Neagu Mircea // Transilvania University, Brasov, Romania, alexandru.oana@unitbv.ro, mircea.neagu@unitbv.ro
In this paper we expose on the dual 1-jet space J1*(R,M^4) the distinguished (d-) Riemannian geometry (in the sense of d-connection, d-torsions, d-curvatures and some gravitational-like and electromagnetic-like geometrical models) for the (t,x)-conformal deformed Berwald-Moor Hamiltonian metric of order four.
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FINSLER GEOMETRY IN GTR IN THE PRESENCE OF A VELOCITY DEPENDENT GAUGE FIELD 2012jfq | Darvas Gyorgy // Symmetrion and Institute for Research Organisation of the Hungarian Academy of Sciences, Budapest, Hungary, darvasg@iif.hu
Fields around their source determine a curved geometry. Velocity dependent phenomena in these fields involve a curvature tensor, whose elements depend on the value and the direction of the velocity of the source of the interaction gauge field, as observed from the reference frame of the matter field. This double (space-time and velocity) dependency of the curvature requires the field to follow a Finsler geometry.
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MOCANU’S PARADOX AND QUATERNIONIC TRANSFORMATION AS THE ANSWER 2012jiq | Ahmad Mushfiq // Rajshahi University, Rajshahi, Bangladesh, mushfiqahmad@gmail.com
When two non colinear velocities are added following Lorentz transformation, a Wigner rotation comes into play, and reciprocity requirement is not fulfilled without this rotation: the velocity of B from A is not the negative of the velocity of A from B. Both Mocanu and Ungar have attributed the paradox (failure of reciprocity principle) to both non-commutativity and non-associativity of Einstein' law of addition of velocities. To resolve this problem, Ungar has proposed a "Weak Associative Law" ( a set of corrections) to make Einstein' law of addition commutative and associative. We have shown that the paradox can be resolved without requiring commutativity. We are proposing a hypercomplex Pauli quaternion law of composition of relativistic velocities, which fulfills physical requirements. The proposed hypercomplex law compares well with Einstein's law of addition of velocities and fulfills all relativistic requirements.
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N-COMPLEX ALGEBRA AND ISOTROPIC RELATIVISTIC AND ELECTRODYNAMICS EQUATIONS 2012jgq | Goryunov A.V. // Turan-Astana University, Astana, Kazakhstan, avgor@hotbox.ru
The version of system of hypercomplex numbers called as N-complex numbers is suggested. In this framework a new concept of bicomplex numbers is introduced and algebraic component of bicomplex calculus is thoroughly developed. The elements of algebras of tricomplex and tetracomplex numbers described briefly, as development of bicomplex concepts and notation. The relationship of elements of bicomplex space with elements of the pseudoeuclidean 4-space is investigated. It is shown that all the basic isotropic (light-like) algebraic formulas of the special theory of relativity and electrodynamics can be obtained as a direct consequence of the properties of bicomplex numbers.
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DOES IT POSSIBLE FINSLER GEOMETRIZATION OF THE POLARIZATION OPTICS? 2012jkq | Ovsiyuk E.M., Redkov V.M. // Mozyr State Pedagogical University, Mozyr, Belarus; Institute of Physics, National Academy of Sciences of Belarus, Minsk, Belarus, e.ovsiyuk@mail.ru, v.redkov@dragon.bas-net.by
This paper provides an overview of some of the features of the matrix calculus to analyze the issues of polarization optics. There is reason to believe that methods of Finsler geometry can be of help here. Since the Mueller matrices, acting on real four-dimensional Stokes vectors, are real, then in possible studies of Mueller matrices one can use their parametrization obtained by applying the Dirac basis. The law of multiplication for the elements of the original group is complicated, but it is suitable for analytical study. The explicit form of the determinant of any matrix in this parametrization, provides us with a natural classifying invariant in a variety of the matrices. This parametrization is used to describe the possible classes of Mueller matrices, including the degenerate cases of matrices with zero determinant, described within the structure of semigroups. It turns out that imposing linear relationships on the 16 parameters that are compatible with the group multiplication law, we can specify mostly classes of degenerate matrices with the structure of semigroups. A complete classification of such semigroups of rank 1, 2, 3 is elaborated.
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EXPLORING RESEARCHES ON THE SPACE-TIME EFFECTS OF HYPERBOLIC FIELDS. PRELIMINARY RESULTS 2012jlq | Pavlov D.G., Panchelyuga M.S., Panchelyuga V.A. // Research Institute of Hypercomplex Systems in Geometry and Physics, Fryazino, Russia; Institute of Theoretical and Experimental Biophysics of the Russian Academy of Sciences, Pushchino, Russia, panvic333@yahoo.com
The paper presents preliminary results of experiments designed to search for hyperbolic or H-fields, which, according to the theoretical representations developed in [1-13], should lead to a local modification of the flow of time. As a generator of the H-field, it is used a process of mechanical strike and as a detector, a highly stable quartz generator. The result of the influence of the H-field on the quartz generator has to be a modification of its oscillation parameters. In the experiment, it is discovered a shift of the power spectrum of oscillation of the quartz generator in the moment of the strike, in comparison with the spectrum in control, obtained under the same conditions, but without the strike. Keywords: Finsler geometry, Berwald-Moor metric, hyperbolic fields, quartz generator.
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THEORETICAL ESTIMATION OF EXPERIMENTAL EFFECT OF POWER SPECTRUM SHIFT OF QUARTZ GENERATOR SIGNAL IN VICINITY OF NONSTATIONAR PROCESSES 2012jmq | Kokarev S.S. // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia; RSEC Logos, Yaroslavl, Russia, logos-center@mail.ru
The paper presents preliminary results of experiments designed to search for hyperbolic or H-fields, which, according to the theoretical representations developed in [1-13], should lead to a local modification of the flow of time. As a generator of the H-field, it is used a process of mechanical strike and as a detector, a highly stable quartz generator. The result of the influence of the H-field on the quartz generator has to be a modification of its oscillation parameters. In the experiment, it is discovered a shift of the power spectrum of oscillation of the quartz generator in the moment of the strike, in comparison with the spectrum in control, obtained under the same conditions, but without the strike. Keywords: Finsler geometry, Berwald-Moor metric, hyperbolic fields, quartz generator.
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