The genetic code, 8-dimensional hypercomplex numbers and dyadic shifts

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 unexpected connections of the genetic system with Walsh functions and Hadamard matrices, which are known in theory of noise-immunity coding, digital communication and digital holography. Dyadic-shift decompositions of such genetic matrices lead to sets of sparse matrices. Each of these sets is closed in relation to multiplication and defines relevant algebra 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 used in analyzing genetic phenomena. The statement about existence of the geno-logic code in DNA and epigenetics on the base of the spectral logic of systems of Boolean functions is put forward. Our results show promising ways to develop algebraic-logical biology, in particular, in connection with the logic holography on Walsh functions.Comment: 108 pages, 73 figures, added text, added reference

The genetic code, algebra of projection operators and problems of inherited biological ensembles

This article is devoted to applications of projection operators to simulate phenomenological properties of the molecular-genetic code system. Oblique projection operators are under consideration, which are connected with matrix representations of the genetic coding system in forms of the Rademacher and Hadamard matrices. Evidences are shown that sums of such projectors give abilities for adequate simulations of ensembles of inherited biological phenomena including ensembles of biological cycles, morphogenetic ensembles of phyllotaxis patterns, mirror-symmetric patterns, etc. For such modeling, the author proposes multidimensional vector spaces, whose subspaces are under a selective control (or coding) by means of a set of matrix operators on base of genetic projectors. Development of genetic biomechanics is discussed. The author proposes and describes special systems of multidimensional numbers under names united-hypercomplex numbers, which attracted his attention when he studied genetic systems and genetic matrices. New rules of long nucleotide sequences are described on the base of the proposed notion of tetra-groups of equivalent oligonucleotides. Described results can be used for developing algebraic biology, bio-technical applications and some other fields of science and technology.Comment: 110 pages,82 figure