Biosimilar properties of biquaternion zero divisors by Sergey Kotkovsky

Sergey Yakovlevich Kotkovsky, leading developer of the NeoFill Internet company. He was born in Chernovtsy (Ukraine). Parents were mechanical engineers of a machinery plant. He graduated from high school number 33 with a gold medal. In 1996, he graduated with honours from the Moscow Institute of Physics and Technology with a degree in Elementary Particles.

He worked after graduation under the guidance of professor K.A. Ter-Martirosyan at the Institute of Theoretical and Experimental Physics (Moscow). The works of this period are devoted to the study of heavy mesons and baryons decays. Now he lives and works in Cleveland, USA. He is engaged in the creation of Internet-based systems, including software design, development and integration.

The main research interests include nonlinear electromagnetic field theory, vector algebra, and quaternion algebra and analysis. Among the obtained scientific results are nonlinear Maxwell equations, nullvector algebra, algebra of vector cycles. Some of these results may be related to biological processes and the future theory of biological field. In systems theory, the main interests lie in the study of directed self-organization, system phase transitions, and the development of social systems .

The biquaternions were discovered by Hamilton after the discovery of quaternions, as a complex-valued expansion of the latter. What results have you managed to achieve as a research result in this area?

Hamilton discovered the existence of zero divisors among biquaternions (we call them “nullquaternions”), which remained poorly understood until now. We study the relationship between different types of nullquaternions, in particular, between the so-called “ordinary” nullquaternions and nullvectors; i.e. complex zero-valued three-dimensional vectors. As a result, we discovered some properties, the main of which are expressed by the theorems on nullvector factorization and allelism.

But alleles is a biological term, which means the same gene different forms located in the same regions (loci) of homologous chromosomes that determine a particular trait development direction. How are alleles related to algebraic objects?

There is a property that indicates the similarity between nullvectors and biological genes. This is a nullvector factorization consequence: when multiplied by each other, each of two nullquaternions gives its structural halve to the product. Half of the parent’s genetic information is passed on to the descendant in crossbreeding genetics. In the nullquaternion case, either the left or right structural half acts as such half, depending on the two nullquaternions product order.

In two nullquaternions product, one of the two structural halves of each cofactor is preserved. The latter circumstance indicates a remarkable nullvector algebra similarity with genetics: the nullvector product acts like combination of allelic genes in the chromosome.

Thus, nullquaternionic product is mathematical analogue of a diploid genetic crossing. It is somewhat continuation of S.V. Petukhov’s idea of algebraic foundation of the living nature processes, connecting molecular genetic alphabets and hypercomplex numbers.

Our study of the nullvector algebraic properties, also called isotropic vectors, is also of interest to physics. In particular, superposition of isotropic vectors associated with massless fields serves as the theoretical construction basis of the so-called A.V. Gorunov walking wave simulating massive particle.

How convenient is the matrix language for describing the biquaternions properties discovered by you?

It is known that the biquaternion ring is isomorphic to the second rank complex matrices (Pauli matrices) ring. That is why our results can also be obtained using matrix language. However, the scalar-vector representation used by us is clearer and more useful from a methodological viewpoint.

Let’s pay attention to how our vector terminology corresponds to the matrix one. The biquaternion squared absolute value defined in the matrix representation is the matrix determinant. Nullquaternions correspond to degenerate matrices (whose determinant is zero), vectors correspond to traceless matrices, and nullvectors correspond to degenerate traceless matrices (nilpotent matrices).

We can make a general conclusion that any nullquaternion class is an algebra over the complex number field. Such a statement cannot be made about any biquaternion classes, but only about nullquaternion classes.

The quaternionic scalar-vector description advantage over a matrix one is that in it we deal exclusively with quaternions, while in the case of matrices, it is necessary to select their special subclasses from the latter manually.

What are the prospects for molecular genetics and biophysics in your theory?

When studying nullvector algebra, we found their pronounced similarity to the paired genes pattern of inheritance and its manifestation. This gives us hope for the nullvectors to be used to model the genes themselves. But physically the nullvector describes a plane electromagnetic wave with circular polarization. Based on this analogy, the structure of the nullvector simulating gene indicates the massless nature of the latter, considered as a field object. Such objects emerge in nonlinear field theory as solitons. The nonlinear field proposed for consideration include, in addition to the usual force (energy) interactions also phase and structural interactions. Phase interactions work to synchronize the resonances frequencies and phases. But the development of rigorous theory of biological field is a separate topic.

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