


Volume 9 No 4 (2011)
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Matrix Genetics and Algebraic Properties of the Multi-Level System of Genetic Alphabets
Sergey V. Petoukhov
Abstract
The article is devoted to algebraic properties of the multi-level system of moleculargenetic alphabets. It leads to help solve the problem of algebraic unity of inherited
information systems in living matter. These algebraic properties are revealed by
means of Kronecker families of matrix forms of a presentation of molecular-genetic
alphabets. A family of genetic (8x8)-matrices shows unexpected connections of the
genetic system with Rademacher and Walsh functions and with special Hadamard
matrices which are well-known in theory of noise-immunity coding and digital
communication. Decompositions of such genetic (8x8)-matrices on the basis of the
known principle of dyadic-shifts lead to sets of 8 sparse matrices. Each of these sets
is closed in relation to multiplication and defines a special algebra of 8-dimensional
hypercomplex numbers. Mathematical aspects of these 8-dimensional algebras are
presented in connection with metric vector spaces, the sequency theory by Harmuth
and some methods of spectral analysis. The diversity of known dialects of the genetic
code can be analyzed from the viewpoint of these algebras. Our results are discussed
taking into account the important role of dyadic shifts, hypercomplex numbers and
Hadamard matrices in mathematics, informatics, theoretical physics, etc. These
results testify that living matter has a profound algebraic essence which is
interconnected with 8-dimensional vector spaces. In our opinion these results lead to
a new way of knowledge of living matter in the field of algebraic biology and its
mathematical modeling. The idea of a biological meaning of Kronecker multiplication
of matrices is based on the structure of Punnett squares in the field of Mendelian
genetics. The author believes that the Mendelian laws of independent inheritance of
traits have revealed just the tip of an algebraic iceberg of informational structure of
living matter and that matrix genetics has contributed to the next steps to disclose
this important iceberg.
Keywords
genetic code, alphabet, Punnett squares, dyadic shift, hypercomplex numbers, Hadamard matrices, symmetry
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