Geometric Invariants Under the Möbius Action of the Group SL(2;R)

In this paper we have introduced new invariant geometric objects in the homogeneous spaces of complex, dual and double numbers for the principal group SL(2; ℝ), in the Klein’s Erlangen Program. We have considered the action as the Möbius action and have taken the spaces as the spaces of complex, dual and double numbers. Some new decompositions of SL(2; ℝ) have been used.

Double and dual numbers. SU(2) groups, two-component spinors and generating functions

We explicitly show that the groups of $2 \times 2$ unitary matrices with determinant equal to 1 whose entries are double or dual numbers are homomorphic to ${\rm SO}(2,1)$ or to the group of rigid motions of the Euclidean plane, respectively, and we introduce the corresponding two-component spinors. We show that with the aid of the double numbers we can find generating functions for separable solutions of the Laplace equation in the $(2 + 1)$ Minkowski space, which contain special functions that also appear in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.

The Genetic Code, Algebraic Codes and Double Numbers

The article shows materials to the question about algebraic features of the genetic code and about the dictatorial influence of the DNA and RNA molecules on the whole organism. Presented results testify in favor that the genetic code is an algebraic code related with a wide class of algebraic codes, which are a basis of noise-immune coding of information in communication technologies. Structural features of the genetic systems are associated with hypercomplex double (or hyperbolic) numbers and with bisymmetric doubly stochastic matrices. The received results confirm that represented matrix approaches are effective for modeling genetic phenomena and revealing the interconnections of structures of biological bodies at various levels of their organization. This allows one to think that living organisms are algebraically encoded entities where structures of genetic molecules have the dictatorial influence on inherited structures of the whole organism. New described algebraic approaches and results are discussed.

The Genetic Code, Algebraic Codes and Double Numbers

The article shows materials to the question on algebraic features of the genetic code. Presented results testify in favor that the genetic code is an algebraic code related with a wide class of algebraic codes, which are a basis of noise-immune coding of information in communication technologies. Algebraic features of the genetic code are associated with hypercomplex double (or hyperbolic) numbers. The article also presents data on structural relations of some genetically inherited macrobiological phenomena with double numbers and with their algebraic extentions. The received results confirm that multidimensional numerical systems is effective for modeling and revealing the interconnections of structures of biological bodies at various levels of their organization. This allows one to think that living organisms are algebraically encoded entities.

Idea of common good (el bien general) in the series of etchings “The Disasters of War” (“Los Desastres de la Guerra”) by Francisco Goya (based on the Bohdan and Varvara Khanenko Museum of Art funds)

The ideas of the Enlightenment (first of all the French, with the most famous of its representatives – Jean-Jacques Rousseau, Charles-Louis de Secondat, Baron de La Brède et de Montesquieu and François-Marie Arouet Voltaire) not only influenced the political sphere of the Eighteenth century but also art. Francisco José de Goya y Lucientes (1746-1828) was directly convinced by these ideas: he took a passive part in the Napoleonic wars and was a friend of the prominent representatives of the Spanish Enlightenment. The study aims at analyzing interactions between text and image in the series of etchings of F. Goya “The Disasters of War” and the reception of the idea of «common good» in the etching 71 “Against the common good”. We have chosen several theoretical and methodological tools to deal with narrative and visual sources. Hermeneutics and semiotics belong to the specific methods used in the process of analysis of engravings. Comprehensive approach is determined by the usage of F. Goya both extraverbial and verbal (double numbers of etchings and artionims, ekfrasis) means. The methodological basis of the study is made up wit the principles of complexity, historicism and scientific character. The main methods were iconographic and iconological; empirical, prosopographical, method of synthetic and analytical source criticism; comparative-historical analysis. Probably, Francisco Goya, who also criticized the contemporary obscurantism in Spain (which is especially reflected in the series of etchings “Los Caprichos”), turned to the ideas of the French enlightenment, which gave rise to possibly unconscious reminiscences and allusions in his work. Thus, we are interested mainly how Goya indirectly or even unconsciously borrowed ideas from the Enlightenment movement, which spread rapidly all over Europe. In this case studying direct borrowings from J.-J. Rousseau’s ideas played only minor role.

Three-Polar Space Over the Semi-Field of Double Numbers

Multi-polar space is a generalization of the notion of vector space. In this paper, we deal with a three-polar vector space over a semi-field of double (hyperbolic complex) numbers. We introduce and study operations of addition and multiplication such that they form a commutative ring with unit on the three-polar space

Multidimensional Numbers and the Genomatrices of Hydrogen Bonds

This chapter returns to the kind of numeric genetic matrices, which were considered in Chapter 4-6. This kind of genomatrices is not connected with the degeneracy of the genetic code directly, but it is related to some other structural features of the genetic code systems. The connection of the Kronecker families of such genomatrices with special categories of hypercomplex numbers and with their algebras is demonstrated. Hypercomplex numbers of these two categories are named “matrions of a hyperbolic type” and “matrions of a circular type.” These hypercomplex numbers are a generalization of complex numbers and double numbers. Mathematical properties of these additional categories of algebras are presented. A possible meaning and possible applications of these hypercomplex numbers are discussed. The investigation of these hyperbolic numbers in their connection with the parameters of molecular systems of the genetic code can be considered as a continuation of the Pythagorean approach to understanding natural systems.