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.

The paper deals with the comparative analysis of two methods for local analysis of
noise-like time series. One of the methods, a local fractal analysis by all permutations
combinations method (APM) [1, 2], has already proven its effectivity as an instrument
that allowed authors to reveal a stable frequency structure in time series of the ?-decay
rate fluctuations [3, 4]. The second one is a pairwise expert comparison of the smoothed
histogram shapes [1-2, 5]. The methods share a set of the properties: invariance under
linear transformations (shifts, dilatations, and mirror reflections), the same as invariance
with respect to rearrangement of terms in section of time series that bases calculations of
the APM dimensions and construction of a smoothed histogram. The analysis showed that
74% of the pairs selected fully automatically by the APM are the same that were selected
in course of expert pairwise histogram comparison, while 26% of the ACM selected pairs
were never selected by any expert.

We present a review of works on the algebraic realization of the “one-electron Universe”
concept of Stueckelberg-Wheeler-Feynman. Two different mechanisms of “multiplication”
of copies-particles on a unique World line (WL) are proposed: implicit definition of a
WL by a system of algebraic equations and the light cone equation (LCE) in the process
of detection by an external observer. In both cases, for polynomial and/or rational
parameterization of a WL, there arises the correlated dynamics of two kinds of particles
corresponding to real (R-) or complex congugate (C-) roots of polynomial equations. As
a direct consequence of the Vieta’s formulas, this dynamics turns out to be conservative:
for the set of RC- particles the laws of conservation of total momentum, angular
momentum and (the analogue of) total energy do hold. Satisfaction of the Newton’s
laws and generation of mass of an arbitrary value for a system of two macroscopic
bodies are established. In the model based on the LCE the collective RC-dynamics is
Lorentz-invariant, and the total rest mass is necessarily integer-valued. At great values
of the observer’s proper time, the effect of “coupling” of RC-particles takes place and,
under specific conditions, – the effect of clusterization of pairs. In the case of à rationally
parameterized WL, three distinct types of RC-particles with specific localization and
dynamics can be asymptotically distinguished.

The notion of the dimensionless gravitational charge of any mass is given. Decay of the
gravitation constant on 41 order gives the growing of the curvature radius on 41 order
of the de Sitter vacuum from 10?13cm to 1028cm and change of vacuum density from
the Planckian value to the critical one. Hypothesis of friedmons as dark matter particles
is proposed. Friedmons are stable particles with mass as billion masses of nucleons. The
corresponding hypothetical exact symmetry group S?U (2) is dual to the group SU(2) of
the broken symmetry in frame of the Standard model with stable particles for the other
exact symmetry groups SU(3),U(1). There is proposed also hypothesis of a dissociation
of primordial de Sitter world with the Planckian density to the modern expanding
Universe with the modern critical density. Then the T-duality and S-duality hypothesis
of the dissociation of primordial symmetry group E(8)? ?E (8) into the subgroups
SU(3)?SU(2)?U(1) and the dual subgroups S?U (3)?S?U (2)? ?U (1) is proposed. These
dualities connect the minimal Planckian length 10?33cm with the primordial curvature
radius 10?13cm of the Universe with the Planckian density (5 • 1093g/cm3)and with the
modern curvature radius of the Universe 1028cm. This thesis is corresponds to the possible
connection between the Planckian mass (near 10?5g) and mass of the Universe (near 1056g).
This work is made in frame of the Program of the Russian Academy of Sciences P7 «The
experimental and theoretical investigations of the Solar system objects and the planetary
star systems. Transient and explosive processes in the astrophysics».

VERY SPECIAL RELATIVITY AND FINSLER GEOMETRY 2016jcw | Olga Chashchina (1), Natalya Dudisheva (2), Zurab Silagadze (3) // (1) Ecole Polytechnique, Palaiseau, France; (2) Novosibirsk State University, Novosibirsk, Russia; (3) Budker Institute of Nuclear Physics, Novosibirsk, Russia, chashchina.olga@gmail.com, dudyshevan@mail.ru, z.k.silagadze@inp.nsk.su

Connections between Cohen and Glashow’s very special relativity and a Finsler
generalization of the special relativity is investigated. It is argued that if we stick to
Einstein’s two postulates then the very special relativity in its pure form is unnatural and
only in the context of the Lalan-Alway-Bogoslovsky generalization of special relativity,
based on a Finslerian metric, it makes a perfect sense.

The article deals with polyadic operanions on sets of cubic matrices, all sections for
orientations (i), (j) and (k) of which are nonsingular matrices and at the same time are
simmetric relative to main diagonal as well as secondary diagonal.

SYMMETRIES OF THE PHASE PLANE 2016jbw | V.V. Smolyaninov // IMASH RAS, Moscow, Russia; ITEB RAS, Puschino, Russia
smolian@mail.ru

Phase portraits are widely used in the theory of oscillations of mechanical and other
systems [1] as they give complete and evident geometrical representation of all variety of
movements of dynamic system under various entry conditions. In the present work new
geometrical interpretation of phase portraits of the linear dynamic systems is offered.
According to well-known F. Klein’s program [2] any geometry follows represents as the
theory of geometrical invariants. Constructive working out of this program is reduced
to identification of systems of basic invariants, which generating the metric property
for spatial movements. Earlier we used such approach for the purpose of identification
kinematic invariants of common chronogeometries, and also the special relativity [3-5].
Here we will consider simple dynamic models of classical mechanics.

SYMMETRIES OF QUATERNIONS 2016jaw | V.V. Smolyaninov // IMASH RAS, Moscow, Russia; ITEB RAS, Puschino, Russia
smolian@mail.ru

The modern mathematical definitions of symmetries are reduced to identification of the
appropriate transformation groups. According to such definition, the group algebras
have the symmetries of discrete groups of their basic elements. In a particular, the basic
elements of Hamilton’s quaternion form discrete «quaternionic group» of 8-th order.
There are only 5 discrete groups of 8-th order, therefore is admitted to speak about
existence 5 of types quaternions, one of which is «hamiltonian quaternion», and others
four are «unhamiltonian quaternions». In job the comparative description all of five types
quaternions is given. With the purpose of unification of comparisons the generalized model
of quaternion is entered.

In this paper, we study the properties of biquaternionic divisors of zero («nullquaternions»).
Subalgebra of nullquaternions is closely related to its subclass – «nullvectors», which are complex-valued three-dimension vectors, having zero square. Theorem of nullvector factorization shows that regular nullquaternion can be represented as product of two nullvectors belonging to uniquely defined classes, thus defining the structure of nullquaternion. Theorem of nullvector allelity proves that product of two nullquaternions preserves one of the structure halves of each multiplier. Last circumstance signs for prominent similarity of nullvector algebra with genetics: product of nullvectors is similar to combination of genes in chromosome. We show, that along with nullvectors there exist
«uniform» classes of nullquaternions which are isomorphic to nullvector classes. Regular, uniform nullquaternions and nullvectors represent general classification of nullquaternions
with relation to multiplication.

The paper presents an action of an interacting particles system based on a hyperbolic-spherical-symmetric decision of a wave equation in the Minkowski space-time, space-time analog of the Colomb’s law, and the superposition law. The corresponding motion equations are integro-differential, and their notation according to the Newton’s second law in the relativistic form reveals the dynamic nature of a mass: it becomes a result of a cooperative effect of a part of hyperbolic interaction between the particle and its environment (the Mach’s principle). The author analyses some special cases of the hyperbolic self-action of a solitary world line and interactions between a pair of particles and parallel world lines.

The paper shows that if one knows a Finsler function ?(p; x), and a function S(x), that
is an operation as a function of coordinates, determining a normal congruence of world
lines, and knows an element of proper times along them, one can draw a differential
equation with partial derivatives for a field of proper times T(x) of the normal congruence
of world lines. Interrelation of such sort between the abovementioned ideas is of special
interest, when hypersurfaces of the level T(x)=const are the transversal supersurfaces to
the normal congruence of the world lines determined by the world function SW(x).

The author suggests to interpret time as a field of synchronized proper times of a normal
congruence of world lines. He lays down the conditions allowing to consider a function of a point of the events space as a parameter of evolution. The length of a world line from a normal congruence of extremums is the best suited parameter of evolution, that is, an operation is the same as a point function in the Newtonian mechanics. Whether the world field (the field of the world function or a meatspace at zero-order approximation) and a field of synchronized proper times of a normal congruence of world lines are interrelated? If a field of the proper times can be expressed through a world function only, then hypersurfaces of the world function level are the hypersurfaces of the proper times of normal congruencies of the world lines. A space per se (at any moment of time) is transversal to all the world lines from the normal congruence.

There is the amendment to the paper «Ternary product over a three-dimensional
matrices» (Hypercomplex numbers in geometry and physics, 1 (21), 2014. p. 157-179),
which corrects equations, representing partial cases of the ternary product of unit matrices in the algebra .

The biquaternionic wave equation with vector structural coefficient is investigated . With
use of the theory the generalized functions and scalar potentials the fundamental and
generalized solutions of this equation are constructed at any regular or singular right part. The wave and energy properties of elementary decisions are investigated.

The article is devoted to the new modeling approach to study the role of wave and
vibration processes in genetically inherited organization of living bodies. This approach
is based on matrix analysis and uses the well-known property of matrices for displaying
resonances. Emphasis is placed on systems of resonances in families tensor matrices based
on the tensor product. The concept of inheritance tables eigenvalues of the matrices
of these families is introduced. Their analogies are shown with Punnet’s squares of
Poly-hybrid crosses of organisms under the Mendel’s laws.Matrix analysis gives evidences
in favor of the following: alphabets of the genetic code are alphabets of resonances;
respectively, the genetic code is the code of resonances, and genetic texts, which are based on these alphabets, are written in the language of resonances; alleles of genes, which are represented in Mendel’s laws, can be interpreted as resonances (the eigenvalues of matrices) of some oscillatory systems. The conception of resonance genome is formulated. Ideas of vibrational genetic biomechanics are under development taking into account connections between inherited biological processes and phenomena of vibrational mechanics.

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METRICAL DYNAMICS 2015jcv | S.V. Siparov // Civil Aviation State University, Saint-Petersburg, Russia; NRU of Information Technologies, Mechanics and Optics, Saint-Petersburg, Russia, sergey.siparov@gmail.com

The suggested approach makes it possible to produce a consistent description of motions
of a physical system. It is shown that the concept of force fields defining the systems’
dynamics is equivalent to the choice of the corresponding metric of an anisotropic space,
which is used for the modeling of physical reality and the processes that take place.
The examples from hydrodynamics, electrodynamics, quantum mechanics and theory
of gravitation are discussed. This approach makes it possible to get rid of some known
paradoxes; it can be also used for the further development of the theory.

Since the beginning of the quest of hypercomplex numbers in the late eighteenth century,
many hypercomplex number systems have been proposed but none of them succeeded in
extending the concept of complex numbers to higher dimensions. This paper provides
a definitive solution to this problem by defining the truly hypercomplex numbers of
dimension N 3. The secret lies in the definition of the multiplicative law and its
properties. This law is based on spherical and hyperspherical coordinates. These numbers
which I call spherical and hyperspherical hypercomplex numbers define Abelian groups
over addition and multiplication. Nevertheless, the multiplicative law generally does
not distribute over addition, thus the set of these numbers equipped with addition and
multiplication does not form a mathematical field. However, such numbers are expected
to have a tremendous utility in mathematics and in science in general.

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ON CUBIC MATRICES 2015jav | A.M. Gal’mak // Mogilev State University of Food Technology, Mogilev, Belarus, halm54@mail.ru

The article deals with cubic matrices of tree types: cubic matrices of order n, whose r-th
sections of orientations (i), (j), (k) for any r =1, 2, . . . , n are the same; cubic matrices all elements of which are symmetric both as for the main diagonal and as for the secondary one in each section of any orientation; cubic matrices of a set Cn?n?n(P), that was determined by the author. All cubic matrices involved are similar to each other because
of being symmetric.

The paper presents the ternary generalization of the standard algebra of matrices to the case of spatial (cubic) matrices. In addition to the definition of the ternary
operation itself and to its basic properties, the multy-dimensional versions of the other conventional concepts, operations and mapping such as transposition, unit element, commutativity, associativity and others that are used in the standard algebra of matrices are constructed. The relationship between the constructed ternary operation and the algebra of polynumbersP3is discussed.

Results of local fractal analysis of 329 1-day time series of Pu-239 alpha-decay rate fluctuations by means of all permutations method (APM) [1] are presented. The APManalysis reveals in the time series some steady frequency set. Coincidence of the frequency set with the Earth natural oscillations was demonstrated. Short revue of periods in fluctuations of various processes (physical, chemical, biological) in range 1-120 min described in works of different authors are given. We shown that periods observed in cited papers corresponds to periods, which revealed in present report. Such correspondence leads to conclusion about some common mechanism, which may cause observed periodicity in processes of different nature.

Present paper continues investigation of possible manifestations of hyperbolic field. The paper contains preliminary results of experimental studies of ultrastable quartz generator frequency change in neighborhood of powerful electric discharge. Obtained results show that in moment of discharge appears frequency shift in output signal of quartz generator. Under the same condition but without the discharge the frequency shift is absent. Obtained results may be considered as possible evidence
of hyperbolic field existence.

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Local fractal analysis of noise-like time series by all permutations method 2014jgx | Panchelyuga V.A., Panchelyuga M.S. // 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

In the present work local fractal analysis of non-stationary time series by all permutations method (APM) is developed. APM-method [1] incorporates ideas of method of minimal cover [2] and histograms method [1]. Analysis of histograms method achieves that some periods in noise-like time series can be revealed only by means of the
method and cannot be find out by traditional methods of time series analysis like correlation analysis, spectral analysis, dispersion analysis and so on. Connection between shapes of smoothed histograms constructed on the base of short segments of time series of fluctuations and fractal dimension of the segments is studied. Is shown that fractal dimension posses all main properties of histogram method. On this base a further development of fractal dimension determination algorithm is proposed. This algorithm allows precision determination of fractal dimension by using short (30-60 points) time series segments. This property of APM-method leads to possibility of analysis of non-stationary time series.

It is shown that formalism of linear vector space is inadequate at the metric approach to geometry, when geometry is described completely in terms of the distance functiond, or in terms of the world function σ=d^2/2. Operations of the linear vector space appear to be ambiguous, if they are introduced at the metric approach to geometry.

The comments clarify the essentials and results of the mentioned paper for the theoreticians who are not well aquainted with the experimental technic. It is also
suggested to feel more free while using the notion of «ether», if this helps to demonstrate the heart of the matter. It is also shown that alongside with the
interpretation given by the authors which is based on this notion, it is possible to use the geometrical approach.

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Displacement of light during satellite laser ranging 2014jdx | Ignatenko Yu.V., Ignatenko I.Yu., Tryapitsyn V.N. // Crimean Laser Observatory of the Main Astronomical Observatory of the National Academy of Sciences of Ukraine, Yalta, Republic of Crimea; Federal State Unitary
Enterprise “Russian Metrological Institute of Technical Physics and Radio Engeneering” (VNIIFTRI), Mendeleevo, Russia, igig@vniiftri.ru

This article describes the results of a study of the anomalous deviation of light detected during laser ranging of the Earth artificial satellites. A special technique
developed for construction of a three-dimensional vector of the laser beam deviation based on processing of its projections onto a focal plane of a telescope is
stated. Appropriate equations are derived. A method and results of test measurements of the displacement of light near the surface of the Earth, confirming the
universal nature of this phenomenon, are described. From these results, the conclusion of the motion of light medium, traditionally called the luminiferous ether, at a
rate in magnitude and direction close to but not equal to the velocity of the Earth is derived. Measured the deflection of light from a given direction is the result
of adding the relative velocity of the satellite, velocity of the Earth, and finally light medium speed. This last fact explains the seasonal dependence of the
measurements.

Two known, alternative to each other, forms of presenting the Maxwell electromagnetic equations in a moving uniform medium are discussed. The commonly used Minkowski
approach is based on two tensors; the relationships between them change their form under Lorentz transformations and take the shape of Minkowski equations, depending
upon the 4-velocity of the moving particle in an inertial reference frame. In this approach, the wave equation for the electromagnetic 4-potential has a form which
explicitly involves this 4-velocity vector of the reference frame. Hence, the Minkowski electrodynamics implies the absolute nature of mechanical motion. An
alternative formalism (proposed by Rosen & al.) may be constructed in new variables, when the Maxwell equations are written in terms of a single tensor. This form of
Maxwell equations exhibits symmetry under modified Lorentz transformations in which, everywhere, instead of the vacuum speed of light cone uses the medium speed of
light c*c. Due to this symmetry, the formulation of Maxwell theory in this medium can be considered as invariant under the mechanical motion of the reference frame,
while the transition must follow modified Lorentz formulas. The transition of the Maxwell equations to 4-potential leads to a simple wave equation which does not
contain any additional 4-velocity parameter, so this form of the electrodynamics presumes a relative nature of the mechanical motion; also, this equation describes
waves which propagate in space with light velocity kc, which is invariant under the modified Lorentz formulas. In connection with these two theoretical alternative
schemes, an essential issue must be stressed: it seems reasonable to perform the Poincar´e-Einstein clock synchronization in uniform media with the help of real light
signals influenced by the medium, which leads us to modified Lorentz symmetry. A similar approach is developed for a spin 1/2 particle obeying the Dirac equation in a
uniform medium.

Paradoxical situation has long been formed in the physical sciences. From the middle of the last century, in estimating the value of the general principle of
relativity (GPR), the opinion of the physics community was divided into two oppositely point of view. We will not enumerate all the supporters and opponents of any of
them, and for the sake of brevity, we combine them with the names of their representatives – the greatest physicists of the 20th century. Let’s call the first one –
the point of view of V. Gisnzburg (see, eg, [1]), and the second – the point of view of V. Fock (see, eg, [2, 3]). The first interprets the general theory of
relativity (GTR) of A. Einstein as the most important achievement of physical thought of the 20th century. The second denies the role of GPR as a fundamental physical
principle. Purpose of this work – give an idea of the current state of the question. The true meaning of the principle of relativity is revealed in the introduction of
new geometries to the physical science, that are more general than the geometry of Riemann spaces, serving as the mathematical foundation of general relativity. These
include the geometry of Finsler spaces and its generalizations — the geometry of spaces with an areal metric (see, eg. [11,14]).

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Hyperbolic «statics» in space-time 2014jax | PavlovD.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

With using the concept of matter event as a matter source, concentrated on metric sphere of null radius — light cone of Minkowski space-time, — we derive the analog of
Coulomb’s law for hyperbolic space-time field, universally acting between events of space-time. The collective field, providing interaction of the world lines of a
pair of particles, consists of 3D Coulomb’s part and logarithmic term. We reveal, that the Coulomb’s part is caused by tune balance between causal and geometric
properties of space-time (concordance of a two regularization procedures). Equation of motion (in fact — equation of 4D static) for self-consistent configuration of
two world lines are integro-differential ones. It is shown, that these equations may be rewritten in the form of relativistic second Newton’s law, where the force and
the mass are special parts of universal hyperbolic interaction. We discuss principles and perspectives of using of the device called hyperbolic lense, which is 4D
analog of dielectric lense of standard electrostatic.

On a smooth five-dimensional manifold is considered distribution of codimension 1 with the Finsler metric type Berwald-Moor. We define the intrinsic connection associated with a given metric structure.

The weak gravitational field in curved Finsler space of events Berwald-Moor is considered. From the equations of a geodesic line classical equations of motion of a particle in a limiting case for non-Newtonian three-dimensional space in a gravitational field are given. Linear equations for a metric tensor and their solutions are reduced. The problem on redshift is considered.

For quadratic lagrangian of general type obtained variational equations of gravitational field in Riemannian-Catran space. Structure of irreducible torsion components under plane wave propagation in Riemannian-Catran space is investigated.

The notion of an extended connection closing sub-Finslerian space codimension 1 is introduced. On the zero-curvature distribution of sub-Finslerian space with the Finsler metric an almost contact K¨ahlerian space is obtained.

The symmetry of space time is described by using the so called isometric group. The generators of isometric group are directly connected with the Killing vectors [18]. In this paper, we present an explicit connection between the symmetries in the VSR and isometric group of Finsler space. The Killing vectors in Finsler space are constructed in a systematic way. Further, the solutions of Killing equations are present explicitly in
the isometric symmetry of Finsler spaces. The Killing vectors of Finsler-Berwald space are given and we proved that the 4-dimensional Finsler-Berwald space with constant curvature has 15 independent Killing vectors.

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Spinors, matrix structures, and projective geometry in polarization optics 2013jrz | Elena Ovsiyuk, Olga Veko, Mircea Neagu, Vladimir Balan, Victor Red’kov // Mozyr State Pedagogical University, Mozyr, Belarus; University Transilvania of Brashov, Brashov, Romania;
University Politehnica of Bucharest, Bucharest, Romania; B.I. Stepanov Institute of Physics, NAS of Belarus,
v.redkov@dragon.bas-net.by

The paper discusses the role played by Mueller and Jones formalisms in polarization optics, by addressing the
following aspects: restriction to theSU(2) symmetry, non-relativistic Stokes 3-vectors; Cartan 2-spinors in polarization optics; Jones 4-spinors for partially polarized light; the linear groupSL(4,) and the classification of 1-parametric Mueller matrices; semi-group structure and classification of degenerate Mueller matrices.

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On possible effects of the spinor structures in quantum physics 2013jpz | Elena Ovsiyuk, Olga Veko, Alexandru Oan˘a, Mircea Neagu, Vladimir Balan, Victor Red’kov // Mozyr State Pedagogical University, Mozyr, Belarus; University Transilvania of Bra¸sov, Bra¸sov, Romania;
University Politehnica of Bucharest, Bucharest, Romania; B.I. Stepanov Institute of Physics, NAS of Belarus,
v.redkov@dragon.bas-net.by

The paper discusses the following topics: the spinor structure of space models; the relation between the Dirac–Schwinger quantization rule and the superposition principle in quantum mechanics; the manifestation of spinor space structure in classifying the solutions of the Dirac equation and for the matrix elements which are related to physical quantities; spinors in polarization optics; the Jones formalism for completely and partly polarized light; General Relativity and Riemannian space-time models with spinor structure and tetrad (vierbein) formalism.

Main object of the present paper are polyadic operations on the setÌJ (P). Elements of the set are functions defined on the nonempty setJwith values, which belong to set of all matrixÌ(P)with elements from some ringP. Such polyadic operations for the first time introduce E. Post. He consider the case J={1, . . . , m−1}, where C – field of complex number.

It was argued in earlier work that the four-velocity of a measured quantum particle excitation of a Finslerian quantum field in the tangent space manifold of spacetime is not a suitable Finsler coordinate, whereas the four velocity of the measuring device relative to the vacuum is a suitable Finsler coordinate. Furthermore, in the present work, it is argued that the physical Finsler coordinate for describing the classical motion of a
macroscopic object is the four-velocity of the classical object, which in effect acts as a measuring device measuring the characteristics of the metric field. Specifically, geodesic motion of a macroscopic object in a Finslerian spacetime is considered, where the appropriate physical Finsler coordinate is the four-velocity of the object undergoing geodesic motion. It is also claimed that for a macroscopic object, such as a macroscopic measuring device, consisting of more than Avogadro’s number of atoms, any supposed quantum state is negligibly
small, so that for all practical purposes the object is best described by classical mechanics. It is argued that this and the above follow from a reasonable upper bound on physically possible proper acceleration.

Relativistic quantum mechanics and the properties of Dirac fermions can be generated in a particularly powerful way using two vector spaces which are commutative to each other and which contain identical information. The apparently broken symmetry between the two spaces observed through the quadratic geometry of ordinary space becomes a perfect and unbroken symmetry in the quartic geometry which defines the single physical quantity through which the two spaces can be combined.

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Finsler geometry 2013jlz | Garas’ko G.I. // Electrotechnical Institute of Russia, Moscow, Russia, gri9z@yandex.ru, gri9z.wordpress.com

If before Finsler geometry pretends only on geometrization of classical mechanics than after formulation of Finsler geometry self-sufficiency principle we can speaking that the geometry pretends on whole physics geometrization. From that principle follow field theory equations and electromagnetic field and gravitational field naturally unified in four-dimensional pseudo-Riemannian space and in curved Berwald-Moore space. Energy momentum tensor concerned with conservation laws follows from E. Neter theorem. In weak fields approach from Finsler geometry self-sufficiency principle follows linear field theory equations for several independent fields. In opposite case field equations becomes nonlinear and fields becomes non-independent that leads to superposition principle nonfulfillment. In any Finsler space exists a field or some fields in that space may be supplemented with field, which make sense of action as function of coordinates and analogous to real part
of complex potential on Euclidean plane. We propose name such potential as conformal potential. Nondegenerate polynumbers are finslerian spaces, which are very interesting itself and possibly may use in physics. For any finslerian space is possible to build equation analogous to Schrodinger equation or Klein-Gordon equation. This means that the geometry allows further quantum-mechanical development.

Movement of material point in Newton potential with singularity in origin of coordinates is considered. Differential equation describing dependence of square of orbital velosity on distance from origin of coordinates is obtained and its approximate solution is presented. In the case of large distance from origin of coordinates, square of orbital velosity go to non-zero value, which depends on space-time expansion increment.

The principal features of the generalized equivalence theory (anisotropic geometrodynamics) are discussed. The motivation that led to this approach on the base of the analysis of some inconsistencies present in the foundations of the classical mechanics is given. The results of the use of this approach for the interpretation of some observations on the galactic scale that had no explanations before and for the interpretation of observations without the use of the notion of dark matter are presented.

A physical-geometric interpreting of holomorphic functions over polynumbers variable for a number of holomorphicity classes is investigated with using tangent construction, developed in [5]. It is shown, that any concrete choice of holomorphic function (polynumber potential) defines some field-theoretical model with background space-time of GR together with tensor fields of a various ranges. The question on local causal structure of pseudoRiemannian space-time, obtained by tangent construction in Berwald-Moor space, is investigated in general form. It is shown, that the only two causal types of space-times with signature(+,−,
−,−) and(+,+,−,−) can be generated by tangent construction. The systems of differential equations, defining
polynumber potential for Schwarzschild metric and cosmological FRW-metrics are derived.

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ALGEBRA, GEOMETRY AND PHYSICS OF DOUBLE NUMBERS 2013jfz | 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

The paper presents developed version of talk, given at the seminar, which took place 04.04.2013 in PFUR (Moscow) with sir R. Penrose. Some important analogies of complex algebra and analysis based on double numbers algebra are considered (polar and exponential forms of double numbers, elementary functions over double numbers, linearfractional transformations and hyperbolic spinors, h-holomorphic functions and their properties, h-holomorphic extensions). The second part of the paper is devoted to some physical applications of double numbers algebra (SR and its conformal generalization, the «theory of everything» in Hyperland, extravariational principle). The theory of Hyperland can be treated as low-dimensional version of future more realistic «theory of everything», based on polynumbers algebra Pn.

Russian translation of public lecture of Professor Roger Penrose organized by Bauman Moscow State Technical University and the Research Institute of Hypercomplex Systems in Geometry and Physics. The lecture presents basic introduction to Conformal Cyclic Cosmology

Russian translation of public lecture of Professor Roger Penrose organized by Bauman Moscow State Technical University and the Research Institute of Hypercomplex Systems in Geometry and Physics. The lecture presents some aspects of complex numbers, its physical applications and fundamentals of twistor theory.

The functional space of biquaternions is considered on Minkovskiy space. Here the scalar-vector biquaternions representation is used which was offered by W. Hamilton for quaternions. With introduction of differential operator - a mutual complex gradient (bigradients), which generalize the notion of a gradient on biquaternions space, biquaternionic wave (biwave) equations are considered, their invariance for group of the Lorence-Puancare transformations is proved and their generalized solutions have been obtained. Biquaternionic form of generalized Maxwell-Dirac equation is constructed and its decisions are researched on base of the differential biquaternions algebra. Its generalized decisions are built with use of scalar potential. The new equation for these potential are constructed which unites known equations of quantum mechanics (Klein-Gordon and Schrodinger Eq.). The nonstationary, steady-state and harmonic on time scalar fields and generated by them the spinors and spinors fields in biquaternionic form are constructed.