# 2022 year, volume 26, issue 3 (PDF)

The work belongs to the ﬁeld of theoretical aspects of visual image recognition. The images are treated as ﬁnite sets of points in Euclidean spaces of various dimensions. This is how real images can actually be represented. There is no formal deﬁnition of a visual image yet. The article makes a ﬁrst approximation to this deﬁnition.

**Keywords:** image recognition, visual images.

The paper explores the problem of modeling the motion of a wing in an air ﬂow. For this purpose, we use cellular automata modeling the movement of the air and cellular automaton modeling the wing. The wing has some asymmetrical shape. Cellular automata represent linear motion of particles, but they bypass the wing upon collision with it; moreover, the velocity of particles moving along the longer side of the wing is greater than the velicity of particles on the other side. This leads to appearance of a lifting force. The automaton modelling the wing sees cellular automata from some neighbourhood and calculates the velocity of particles. Based on this, the lift vector is calculated. As a result, the wing changes its coordinates. An explicit formula for calculating velocities in the general case, enough for the wing to lift, was found. A proof of the statement for the simpliﬁed wing proﬁle was also presented. An example showing that the velocity calculated by the explicit formula decreases with a small increase in the angle of attack was given.

**Keywords:** automaton modelling of wing aerodynamics, homogeneous structures with inputs, cellular automaton, lifting force.

The paper considers multidimensional convolutional schemes in the McCulloch-Pitts basis. It is shown that the considered schemes can be implemented by a scheme from the a priori and dynamic parts, in which the calculations in the a priori part are independent of the input data. In this case, the a priori and dynamic parts have a nonlinear depth equal to \[ 2 \].

**Keywords:** convolutional neural network, neural scheme, nonlinear complexity, McCulloch-Pitts model.

We introduce a recurrent structure (spatially) on residual networks, which can improve the performance of the network while saving parameters. We investigate the behaviour of recurrent structures in residual networks based on Riemannian manifolds, introducing curvature as a metric for neural networks. We also experimentally verify that the gain due to the recurrent structure is related to the curvature, and demonstrate the generality of the recurrent structure as a method to improve the performance of the network.

**Keywords:** Neural Netrowks, Riemannian geometry, Recurrent structures, Manifold, Transformers.

The description of new continuum structures of closed automata classes.

**Keywords:** continuum structures, closed classes, automata.

We consider exact attribute-efficient learning of functions from Post closed classes using comparation queries and obtain bounds on learning complexity.

**Keywords:** exact learning, attribute-efficient learning, Post lattice of closed classes, comparation queries.

In this paper,volume circuits are considered, which are the embeddings of circuits of functional elements in space. The class \[ T_{\mathrm{near}} \] of circuits where the outputs are located side by side was considered. For this class, the lower and upper estimates of the potential are obtained. Potential is a measure of power equal to the number of circuit elements that produce one on a given input. In particular, it is shown that for Boolean operators with \[ n \] inputs and \[ m \] outputs, the order of the Shannon function for the \[ T_{\mathrm{near}} \] circuit class is \[ \Theta\left(\frac{m}{n} \cdot {\min}^{1/3}(m, 2^{n/2}) \cdot 2^{n/3} \right) \] for \[ m \ge n \], \[ \log_2(m) = o(2^n), n \rightarrow \infty \].

**Keywords:** circuits from functional elements, volume circuits, circuit power, potential.

The present paper investigates the algorithmic complexity of ﬁnding a family of simple circuits passing every vertice of a digraph with semidegree bound \[ 2 \]. The problem is considered in two variants: as a search and as an optimization problem. It proves to be polinomially solvable in both variants, subsequently an algorithm using time \[ O(n^3) \] and, for a particular formulation of the problem, an algorithm using time \[ O(n^2) \] are suggested where n is the number of the digraph’s vertices.

**Keywords:** digraphs, simple circuits, search problems, optimization, \[P\] class, polynomail solvability.