"Hypercomplex Numbers in geometry and Physics" 1 (22), Vol 12, 2015
j022

Content of Issue

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.

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SPHERICAL AND HYPERSPHERICAL HYPERCOMPLEX NUMBERS MERGING NUMBERS AND VECTORS INTO JUST ONE MATHEMATICAL ENTITY
2015jbv | Redouane Bouhennache  // Independent Exploration Geophysical Engineer/Geophysicist, Beni-Guecha Centre, 43019 Wilaya de Mila, Algeria, redouane.bouhennache@outlook.com

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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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.

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TENSOR PRODUCT OF MATRICES IN STUDYING THE ORGANISM AS A GENETIC SYSTEM OF RESONANCES
2015jdv | S.V. Petoukhov  // Institutes for Machines Science, RAS, Moscow, Russia, spetoukhov@gmail.com

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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THE AMENDMENT TO THE PAPER «TERNARY PRODUCT OVER A THREE-DIMENSIONAL MATRICES»
2015jfv | A.V. Lapshin  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, lavexander@mail.ru

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 .

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