Sequences of Groups, Hypergroups and Automata of Linear Ordinary Differential Operators

Jan Chvalina,Bedřich Smetana,David Staněk,Michal Novák

doi:10.3390/math9040319

Jan Chvalina, Bedřich Smetana + Show 2 more

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Journal: Mathematics	Publication Date: Feb 5, 2021
Citations: 5	License type: CC BY 4.0

Affiliation: Brno University of Technology, University of Defence

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The main objective of our paper is to focus on the study of sequences (finite or countable) of groups and hypergroups of linear differential operators of decreasing orders. By using a suitable ordering or preordering of groups linear differential operators we construct hypercompositional structures of linear differential operators. Moreover, we construct actions of groups of differential operators on rings of polynomials of one real variable including diagrams of actions–considered as special automata. Finally, we obtain sequences of hypergroups and automata. The examples, we choose to explain our theoretical results with, fall within the theory of artificial neurons and infinite cyclic groups.

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IntroductionThis paper discusses sequences of groups, hypergroups and automata of linear differential operators
This paper discusses sequences of groups, hypergroups and automata of linear differential operators. It is based on the algebraic approach to the study of linear ordinary differential equations
Its roots lie in the work of Otakar Borůvka, a Czech mathematician, who tied the algebraic, geometrical and topological approaches, and his successor, František Neuman, who advocated the algebraic approach in his book [1]

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This paper discusses sequences of groups, hypergroups and automata of linear differential operators. It is based on the algebraic approach to the study of linear ordinary differential equations. Its roots lie in the work of Otakar Borůvka, a Czech mathematician, who tied the algebraic, geometrical and topological approaches, and his successor, František Neuman, who advocated the algebraic approach in his book [1] Both of them (and their students) used the classical group theory in their considerations. The use of hypercompositional generalizations has been tested in the automata theory, where it has brought several interesting results; see, e.g., [5,6,7,8] This approach is not the only possible one. Investigations of differential operators by means of orthognal polynomials, see, e.g., [9,10]

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