Biology – Quantitative Biology – Other Quantitative Biology
Scientific paper
2011-02-17
Biology
Quantitative Biology
Other Quantitative Biology
80 pages, 54 figures, added sections 17 and 18, added references, corrected typos
Scientific paper
Matrix forms of the representation of the multi-level system of molecular-genetic alphabets have revealed algebraic properties of this system. Families of genetic (4*4)- and (8*8)-matrices show an unexpected connections of the genetic system with functions by Rademacher and Walsh and with Hadamard matrices. Dyadic-shift decompositions of such genetic matrices lead to relevant algebras of hypercomplex numbers. It is shown that genetic Hadamard matrices are identical to matrix representations of Hamilton quaternions and its complexification in the case of unit coordinates. The diversity of known dialects of the genetic code is analyzed from the viewpoint of the genetic algebras. An algebraic analogy with Punnett squares for inherited traits is shown. Our results are discussed taking into account the important role of dyadic shifts, Hadamard matrices, fourth roots of unity, Hamilton quaternions and other hypercomplex numbers in mathematics, informatics, physics, etc. These results testify that living matter possesses a profound algebraic essence. They show new promising ways to develop algebraic biology.
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