"Hypercomplex Numbers in geometry and Physics" 1 (24), Vol 13, 2016
Content of Issue
SYMMETRIES OF QUATERNIONS
2016jaw | V.V. Smolyaninov // IMASH RAS, Moscow, Russia; ITEB RAS, Puschino, Russia
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.
SYMMETRIES OF THE PHASE PLANE
2016jbw | V.V. Smolyaninov // IMASH RAS, Moscow, Russia; ITEB RAS, Puschino, Russia
Phase portraits are widely used in the theory of oscillations of mechanical and other
systems  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  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.
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, email@example.com, firstname.lastname@example.org, email@example.com
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.
POLYADIC GROUPS OF THE CUBIC MATRICES
2016jcw | A.M. Gal’mak // Mogilev State University of Food Technology, Mogilev, Belarus
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.
ON THE HYPOTHESIS OF FRIEDMONS AS DARK MATTER PARTICLES
2016jew | R.F. Polishchuk // P.N. Lebedev Physics Inst. of Russian Academy of Sciences, Astrospace Center, Moscow, Russia, firstname.lastname@example.org
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».
FIELD AND PARTICLELIKE STRUCTURES ON A UNIQUE WORLD LINE
2016jfw | V.V. Kassandrov // Institute of Gravitation and Cosmology,
Peoples’ Friendship University of Russia, Moscow, Russia, email@example.com
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 ŕ rationally
parameterized WL, three distinct types of RC-particles with specific localization and
dynamics can be asymptotically distinguished.
COMPARATIVE STUDY OF LOCAL FRACTAL ANALYSIS BY ALL PERMUTATIONS METHOD AND PAIRWISE EXPERT HISTOGRAMS COMPARISON ...
2016jgw | V.A. Panchelyuga, M.S. Panchelyuga, V.A. Kolombet, at all. // Research Institute of Hypercomplex Systems in Geometry and Physics, Fryazino, Russia; Institute of Theoretical and Experimental Biophysics RAS, Pushchino, Russia; Chemical and Nuclear Engineering Department, Polytechnic University of Valencia, Spain, firstname.lastname@example.org
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.