遗传密码、8维超复数和并矢位移
The genetic code, 8-dimensional hypercomplex numbers and dyadic shifts
摘要：
矩阵形式的表示的多层次系统的分子遗传字母揭示了这个系统的代数性质。遗传（4*4）-和（8*8）-矩阵的家族显示遗传系统与Walsh函数和Hadamard矩阵的意想不到的联系，这在抗噪编码、数字通信和数字全息理论中是已知的。这样的遗传矩阵的二进移位分解导致稀疏矩阵的集合。这些集合中的每一个都是关于乘法的封闭的，并且定义了超复数的相关代数。它表明，遗传阿达玛矩阵是相同的矩阵表示的Hamilton四元数及其复杂化的情况下，单位坐标。从遗传代数的角度分析了遗传密码已知方言的多样性。一个代数类比与Punnett平方遗传性状。我们的研究结果用于分析遗传现象。基于布尔函数系的谱逻辑，提出了DNA和表观遗传学中基因逻辑码存在的命题。我们的研究结果表明，有前途的方法来发展代数逻辑生物学，特别是在与沃尔什函数的逻辑全息。
英文摘要：
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.
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