This paper deals with the application of the isotopic field-charge spin theory to the electromagnetic interaction. First there is derived a modified Dirac equation in the presence of a velocity dependent gauge field and isotopic field charges (namely Coulomb and Lorentz type electric charges, as well as gravitational and inertial masses). This equation is compared with the classical Dirac equation and there are discussed the consequences [6, 34, 35, 37]. There is shown that since the presence of isotopic field-charges would distort the Lorentz invariance of the equation, there is a transformation, which restores the invariance, in accordance with the conservation of the isotopic field-charge spin [8]. It is based on the determination of the Fìí field tensor adapted to the above conditions. The presence of the kinetic gauge field makes impossible to assume a flat electromagnetic interaction field. The connection field, which determines the curvature, is derived from the covariant derivative of the kinetic (velocity dependent) gauge field. In this case, there appears a velocity dependent metric, what involves a (velocity arrowed) direction-dependent, that means, Finsler geometry [11, 14]. The option of such a «theory of the electrons» (with the words of Dirac) was shown in the extension of his theory in [23]. This paper is an attempt to a further extension.

If velocities u and v add up to give w. The three velocities form a triangle. The same velocities, but in the opposite direction, -v and -u should add up to give -w. Isotropy of space requires that the reversal of direction should reverse the order of addition - -v should come before -u. Lorentz Einstein addition does not fulfill this requirement and Wigner rotation in invoked to correct it. Reciprocal symmetric transformation, we are proposing, maintains the isotropy of space and Wigner rotation is not needed.

It is shown that the space-time geometry should be formulated in terms of the world function, because only description in terms of world function admits one to recognize similar geometrical objects in regions of the space-time geometry with different geometries. The Berwald-Moor geometry formulated in terms of the world function appears to be multivariant geometry, which hardly can be used as a space-time geometry, because in this geometry the world lines wobbling of free particles differs from the real wobbling.

Concepts of bicomplex function and its differential operators in bicomplex space are considered. Interrelations of differential operators in bicomplex space and differential operators in 4-space are obtained. Thus the possibility of calculation of derivatives of bicomplex function on 4-space variables is achieved. As result, the main differential isotropic equations of theory of relativity and electrodynamics are obtained as direct consequence of related bicomplex algebraic formulas of preceded paper.

A brief overview of the basics of the theory of wave equations of elementary particles in the presence of external gravitational field, described as a pseudo-Riemannian structure of space-time, is given. Covariant generalization of the wave equations, set in Minkowski space, are presented for bosons and fermions equally, is presented as the result of a single tetrad recipe by Tetrode-Weyl-Fock-Ivanenko, based on representations of the Lorentz group. The Lorentz group plays a unifying role in describing the fields of all particles (with different spins, massive and massless) in the flat and in curved space-time. The difference lies in the fact that in flat space Lorentz group acts as a global symmetry of the wave equations; and in a pseudo-Riemannian space, it plays a role of local symmetry group (dependent on coordinates). Particular attention is given to the Dirac and Maxwell fields. Because any new theory of physical space-time can be expected to cover already developed and proven models, the question naturally arises: for what should be replaced the method of describing the interactions of elementary particles with a pseudo-Rimannian geometric background, if the space-time endowed with a Finsler structure.The answer to this question, if possible, should be fairly universal and independent of the magnitude of the spin of a particle or its mass. The general answer to this question would provide us with the better understanding what we can expect in physics from the use of Finsler geometry, in the most radical aspect as a basic new geometry of physical space-time.One can also put a more particular question: what effective physical media can be described by using a generalized Maxwell electrodynamics in space-time with Finsler geometry.

In this paper we describe the groups of local transformations of coordinates which preserve unchanged on tangent bundles the two and three dimensional Berwald-Mo´or metrics. Some algebraic properties of these groups are studied. Finally, we suggest the possible structure of these transformations in the general n-dimensional case.

I draw on my earlier work to review various aspects of the differential geometry of a Finsler-spacetime tangent bundle, all based on the possible existence of a physical upper bound on proper acceleration. In particular, the bundle connection and associated differential geometric fields are calculated for a Finsler-spacetime tangent bundle particularized for the case of a statioinary measuring device.

The paper gives a generalization of exponential representation of nondegenerate polynumbers, which are named as ladder representation. It given on the base of hypercomplex numbers H4. The representation arise an iterative process, which can be finite or infinite. Also a new approach to understanding of notions of length and angle in polynumber spaces are proposed.

English:

Russian:

SOME PROBLEMS OF MATHEMATICAL PHYSICS IN POLYNUMBERS FIELD THEORY 2012jnq | 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-center@mail.ru

Some mathematical aspects of the invariant scalar operator În, that appears in polynumbers field theory, are considered together with some characteristic properties of its kernel. Concrete expressions for Berwald-Moor metrics and operator On in isotropic cylindrical, non-isotropic cylindrical and general spherical coordinate systems are derived in case n=3. Part of the results is expressed in terms of special functions, which are hyperbolic analogies of trigonometric ones, spherical harmonics and Legendre polynomials. General kind of radial part of n is calculated for arbitrary În. The problem of finding of hyperbolic field, generated by homogeneously charged hyperbolic sphere, is solved. We show, that there is no separable solutions to hyperbolic wave equation with cylindrical symmetry.

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.

English:

Russian:

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.

English:

Russian:

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.

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.

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.

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.

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.

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.

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.

English:

Russian:

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

Conventional form of the special relativity theory formulates the theory in an unaccomplished form. The dynamic equations of the particle motion are written in accordance with the relativity principles, whereas the particle state is described in the nonrelativistic form. Ignoring the nonrelativistic concept of particle state, one succeeds to construct an uniform formalism for description of deterministic and indeterministic particles, which leads to a necessity of multivariant space-time geometry. The quantum principles are founded by existence of the multivariant space-time geometry and lose the role of prime physical principles. Skeleton conception of the elementary particles realizes relativistic description of the particle state, which appears to be adequate in the case of discrete and multivatiant space-time geometry. The skeleton conception accomplishes transition from nonrelativistic physics to the relativistic one and realizes complete geometrization of physics.

In Finsler geometry a Finsler coordinate is a coordinate in the tangent space manifold of a given base manifold. As such it has been given various definitions in the relativity and field theory literature and often even remains undefined physically. Physically meaningful coordinates of a point in the tangent bundle of spacetime are the spacetime and fourvelocity coordinates of the measuring device. It is here emphasized that the four-velocity of the measuring device need not be the same as the four-velocity of the measured object, either classically or quantum mechanically. The four-velocity of a measured particle excitation of a Finslerian quantum field in the tangent space manifold of spacetime is not a suitable physical Finsler coordinate. The role of the Finsler coordinate is elaborated in a detailed example involving a Finslerian quantum field and associated microcausality.

This article presents some results of investigation of the multi-level system of moleculargenetic alphabets by means of matrix methods from theory of noise-immunity coding. These studies have revealed links of the system of alphabets with some systems of hypercomplex numbers (Hamilton quaternions and Cockle split-quaternions and their extensions), Kronecker families of matrices, orthogonal systems of Rademacher functions and Walsh functions, Hadamard matrices etc. Structural parallels are shown between molecular-genetic alphabets and a system of inheritance of traits in holistic organisms, which obeys the Mendel laws and which is reflected in genetic Punnett squares. The system of molecular-genetic alphabets, common to all living organisms, possesses algebraic properties which lead to a new - algebraic - way of cognition of living matter. This cognition is related with development of algebraic biology associated with hypercomplex numbers. Living matter, providing the transmission of hereditary information in the chain of generations, is presented as information entity deeply algebraic in its nature.

The nilpotent version of the Dirac equation can be constructed on the basis of the algebra of a double vector space or complexified double quaternions. This algebra is isomorphic to the standard gamma matrix algebra, with 64 units which can be produced by just 5 generators. The H4 algebra used in the Berwald-Moor metric is a distinct subalgebra of this 64-part algebra. The creation of the 5 generators requires the rotation symmetry of one of the two component vector spaces to be preserved while the other is broken. It is convenient to identify the respective spaces as an observable real space and an unobservable vacuum space, with corresponding physical properties. In combination the 5 generators produce a nilpotent structure which can be identified as a fermionic wavefunction or solution of the Dirac equation. The spinors required to generate the 4 components of the wavefunction can be derived from first principles and have exactly the same form as the four components of the Berwald-Moor metric. They also incorporate the units of the H4 algebra in an identical way. The spinors produce a zero product which can be interpreted in terms of a fermionic singularity arising from the distortion introduced into the vacuum (or spinor) space by the application of a nilpotent condition.

This article presents some results of investigation of the multi-level system of moleculargenetic alphabets by means of matrix methods from theory of noise-immunity coding. These studies have revealed links of the system of alphabets with some systems of hypercomplex numbers (Hamilton quaternions and Cockle split-quaternions and their extensions), Kronecker families of matrices, orthogonal systems of Rademacher functions and Walsh functions, Hadamard matrices etc. Structural parallels are shown between molecular-genetic alphabets and a system of inheritance of traits in holistic organisms, which obeys the Mendel laws and which is reflected in genetic Punnett squares. The system of molecular-genetic alphabets, common to all living organisms, possesses algebraic properties which lead to a new - algebraic - way of cognition of living matter. This cognition is related with development of algebraic biology associated with hypercomplex numbers. Living matter, providing the transmission of hereditary information in the chain of generations, is presented as information entity deeply algebraic in its nature.

Special relativistic theory in its traditional form formulates theory as incomplete. Particle movement dynamic equation is fixed in accordance with relativistic theory principles while particle condition is fixed in non-relativistic form. Ignoring the non-relativistic idea of particle condition we manage to construct a single formal description for determinate and nondeterminate particles which leads to the necessity of multivariant spacetime geometry. Quantum principles are based on multivariant geometry and lose role of the first physical principles. The frame concept of elementary particles gives relativistic description of particle condition which turns out to be applicable for the case of discrete and multivariant spacetime geometry. The frame concept finishes transition from nonrelativistic physics to relativistic one and realizes complete geometrization of physics.

The paper is a brief review of results on the theory of differentiable functions of polynumbers variable Pn → Pn and of its applications. We define derivative of a function of polynumbers variable, basing on special classification of degenerated (i. e. irreversible) polynumbers and on the theorem stating general form of R-linear mapping Pn → Pn. Then we define holomorphic function of polynumbers variable as subclass of differentiable functions by the set of differential conditions (polynumbers analog of Cauchy-Riemannian conditions), which in isotropic basis have the form: kdf = 0, (k = 1, . . . , n−1) where kd= Ckd, C - conjugation in algebra Pn. Some generalized classes of holomorphic functions Gnka1 ,ka2 ,...,kar are defined by monomic differential equations, which can be classified by the set of vectors of non-negative integer n-dimensional lattice Zn+ . The question of holomorphic continuation of some smooth function from submanifolds of Pn to Pn is discussed. We derive polynumbers version of Cauchy theorem and Cauchy integral formulae together with possible multidimensional generalization the first one. Using symmetric Berwald-Moor form we develop symmetric analog of differential forms calculus (Symmetric product, Hodge star and external differential). We analyze transformation properties of derivatives of scalar polynumbers functions and of those geometrical objects, that can be constructed from these derivatives. In particular, we construct real scalar invariants, appropriate for Lagrangian formalism in polynumbers field theory. Basing on supports algebra we formulate tangent construction, playing important role in physical interpreting of polynumbers field theory. The formula for Levi-Civita connections coefficients concordant with Berwald-Moore form and formula for volume form based on n-root metric of Finsler type in even dimensions are derived. Also we consider some deformational aspects of smooth function of polynumbers variable and prove the statement, that any R-algebra can be embedded into space of bilinear forms over Pn. The paper can be treated as preliminary sketch of general theory of functions of polynumbers variable (TFPV).

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.

English:

Russian:

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.

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.

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.

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.

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.

English:

Russian:

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.

English:

Russian:

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.

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.

English:

Russian:

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.

In the authors paper mentioned in the title of this abstract and published in the Journal «Hypercomplex Numbers in Geometry and Physics», 2 (12), Vol.6, 2009, Pp. 92-137, the statement was formulated that composite idempotents of the Clifford algebra Cl3 of three dimensional Euclidean space generate nonminimal one-sided ideals of Cl3. The amendment presented here cancels this statement: one can prove that all one-sided ideals generated in Cl3 both by composite idempotents and by simple ones are always minimal. Fortunately, the proofs of the rest of results presented in that paper are not affected by this local circumstance and therefore do not fall; as a consequence, all these results remain true.

Mandelbrot set and filled-in Julia sets in double numbers plane have been found numerically. We discuss differences between our definition of these sets and the definition of Artzy. A condition for these two different types of sets to coincide have been found. We show also that our definition allows graphycally interesting escaping time diagramms to be constructed in the double numbers plane, in contrast to the Artzys definition.

Limitations of the vector, tensor and Dirac calculi are illustrated to motivate the Kaehler calculus of integrands, which replaces all three of them and which we introduce in three steps. In a first step, we present the basics of the underlying Clifford algebra for that calculus, algebra valid for Euclidean and pseudo-Euclidean vector spaces of arbitrary dimension. The usual vector algebra is shown to be a corrupted form of Clifford algebra, corruption specific to dimension three and non-existing for other dimensions. The Clifford product is constituted by the sum of the exterior and interior products if at least one of the factors is a vector. Grossly speaking, these products play the role of the vector and scalar products of three dimensions, while generalizing them. It thus contains exterior algebra. As an intermediate step towards the Kaehler calculus, we briefly give the fundamentals of Cartans exterior calculus of scalar-valued differential forms, here viewed as ordinary scalar-valued integrands in multiple integrals. We also make a brief incursion into the exterior calculus of vector-valued differential forms, which is the moving frame version of differential geometry. We show the basics of the Kaehler calculus of differential forms. It is to the exterior calculus what Clifford algebra is to exterior algebra. Because of time and complexity constraints, we limit ourselves to scalar-valued differential forms, which is sufficient for relativistic quantum mechanics with electromagnetic coupling. In using this calculus, the problem with negative energy-solutions does not arise

English:

Russian:

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.

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. Cohomologies of wrap groups and their structure are investigated. Sheaves of wrap groups are constructed and studied. Moreover, twisted cohomologies and sheaves over quaternions and octonions are investigated as well.

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.

English:

Russian:

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.

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.

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.

By analogy with the theory of harmonic fields on the complex plane the theory of wave fields on the plane of double variable is developed. The hyperbolic analogies of point-like sources, curls, source-curls and their multipoles generalizations are constructed. Some important physical aspects of the theory together with possible generalizations on higher polynumber dimensions are discussed.

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

We consider complex-differentiable functions of double variable and their essential properties analogical to the properties of functions of standard complex variable: Cauchy theorem and Cauchy formula, hyperbolic harmonicity, general properties of h-conformal transformations and such transformations, yielded by elementary functions. The question on applications of h-conformal transformations for solving of 2-dimensional problems of mathematical physics is discussed.

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.

The set of the various poly linear forms that can be constructed in the spaces of the non-degenerate poly numbers contains the linear invariant form closely related to the notion of real part of the non-degenerate poly number and to the time coordinate.