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

Abstract

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.

Construction of an Infinite Cyclic Group Formed by Artificial Differential Neurons

“…In accordance with our previous papers [1,[7][8][9], we regard the above mentioned artificial neurons such that inputs x i and weights w i will be functions of argument t belonging into a linearly ordered (tempus) set T with the least element 0. As the index set we use the interval of real numbers [1, ∞) = {x ∈ R; 1 ≤ x}, where R denotes the set of all real numbers.…”

“…It is to be noted, that the used concept of investigated neurons is in a certain sense motivated by ordinary differential operators forming of left-hand sides of corresponding differential equations, see, e.g., [13,14].…”

“…The bodies of artificial neurons compute the sum of the weighted inputs and bias and "process" this sum with a transfer function, cf. [1][2][3][4][5][6][7][8][9][10].…”

“…, x n (t)) is the vector of time dependent (or time varying) input functions, b is the bias and ϕ is the activation function. The use of time varying functions as weights and inputs is a certain generalization of the classical concept of artificial neurons from the work of Warren McCulloch and Walter Pitts (1943); see also [1][2][3][4][5][6][7][8][9][10] and references mentioned therein.…”

State Machines and Hypergroups

“…There followed more papers by the same author and Ch. Massouros, e.g., [13][14][15][16][17][18][19][20][21], as well as other researchers such as J. Chvalina [22][23][24][25][26][27][28], L. Chvalinová [22], Š. Hošková-Mayerová [24,25], M. Novák [26][27][28][29][30][31][32], S. Křehlík [26,27,[29][30][31]33], M.M. Zahedi [34], M. Ghorani [34,35], D. Heidari and S. Doostali [36], R.A. Borzooei et al [37].…”