Acta et Commentationes Exact and Natural Sciences

Mathematical modelling of the immune response to infectious diseases with the influence of environmental factors

2025-02-11T21:06:52+00:00

The mathematical model of the immune response to infectious diseases with the influences of environmental factors is investigated. The conditions for the existence and uniqueness of the solution to the mathematical model for t>0 have been established. Stationary solutions have been identified, along with the conditions for their existence and asymptotic stability. The results are illustrated using a model example.

The problem of the center for cubic differential systems with two affine non-parallel invariant straight lines of total multiplicity three

2025-02-11T21:06:52+00:00

In this paper, we show that a center-focus critical point of cubic differential systems with two affine non-parallel invariant straight lines of total multiplicity three is a center type if and only if the first five Lyapunov quantities vanish.

Center problem for quartic differential systems with an affine invariant straight line of maximal multiplicity

2025-02-11T21:06:52+00:00

In this paper the quartic differential systems with a center-focus critical point and non-degenerate infinity are examined. We show that in this family the maximal multiplicity of an affine invariant straight line is six. Modulo the affine transformation and time rescaling the classes of systems with an affine invariant straight line of multiplicity two, three,..., six are determined. In the cases when the quartic systems has an affine invariant straight line of maximal multiplicity the problem of the center is solved.

On the symbol of singular operators in the case of contour with corner points

2025-02-11T21:06:51+00:00

This paper proposes a method for constructing a symbol for singular integral operators in the case of a piecewise Lyapunov contour. The definition of the symbol function involves numbers that characterize the space in which the research is being carried out, as well as the values of the corner points of the contour, which makes it possible to obtain formulas for calculating the essential norms of singular operators and conditions for the solvability of singular equations with a shift and complex conjugation. In obtaining these results, we will essentially rely on the well-known results of I. Gelfand concerning maximal ideals of commutative Banach algebras \cite{GRS}. In the absence of corner points on the integration contour, the results of this work are consistent with the results from [1].

On computation of the ordinary Hilbert series for Sibirsky graded algebras of differential system s(3,5)

2025-02-11T21:06:51+00:00

The generalized and ordinary Hilbert series for Sibirsky graded algebras of comitants and invariants of autonomous polynomial differential systems are of particular importance for some problems of qualitative theory of differential systems. In the \linebreak Republic of Moldova the computation of these series have their beginnings in the works of Professor M. N. Popa and his disciples. But the construction of these series for some complicated differential systems encounters insurmountable computational difficulties, especially, for the generalized Hilbert series, from which the ordinary Hilbert series can be easily obtained. In this paper, it is shown how the adaptation of Molien's formula address to the mentioned problem to overcome the enormous calculations, an ordinary Hilbert series were obtained for Sibirsky graded algebras of comitants and invariants for the differential system $s(3,5)$.

Stability conditions of unperturbed motion governed by the ternary differential system of Lyapunov-Darboux type with nonlinearities of fifth degree

2025-02-16T21:07:41+00:00

In this paper, there was studied Lyapunov stability of the unperturbed motion for the ternary differential system with nonlinearities of fifth degree on a center-affine variety. The Lyapunov series was constructed and the stability conditions of the unperturbed motion governed by this system were determined in the critical case.

On a method of constructing topological quasigroups obeying certain laws

2025-02-16T21:07:41+00:00

A new method of constructing non-associative topological quasigroups obeying certain laws is given. Also, in this paper we research T-quasigroups with Abel-Grassmann identity (ab)•c=(cb)•a.

Affine invariant conditions for a class of differential polynomial cubic systems

2025-02-16T21:07:41+00:00

In this article the affine invariant criteria constructed in terms of algebraic polynomials with coefficients $\tilde a \in \mathcal R^{20}$ for a class of cubic systems are established. We are focused on non-degenerate real cubic systems with 7 invariant straight lines, considering the line at infinity and their multiplicities and possesing four real singularities at infinity. Additionally, the only configurations of the type $(3, 3)$ of mentioned systems are considered and we denote this class by $CSL_{(3,3)}^{4r\infty}$. In \cite{Buj-Sch-Vul-EJDE-2021} the existence of exactly 14 configurations of invariant straight lines for systems in $CSL_{(3,3)}^{4r\infty}$ was proved. Here we complete this classification by determining necessary and sufficient conditions for the realization of each one of the 14 configurations in terms of affine invariant polynomials.

User perception analysis of the developed AR applications: satisfaction and development directions

2025-02-16T21:07:40+00:00

This article explores the application of Augmented Reality (AR) in education, specifically focusing on the use of AR-based flashcards to support deep learning of mathematical concepts (geometry, Pi number) and vocabulary acquisition (metaphorical terms). AR flashcards offer an innovative solution by integrating dynamic, multimedia-rich content, which enhances understanding and engagement. Prototypes were tested with various user groups, including middle school and university students, who provided valuable feedback through surveys. The SWOT analysis revealed strengths such as clarity and usefulness, particularly in subjects like mathematics and biology, but also identified areas for improvement, such as technical issues and interface design. Based on user input, the design of animal-themed markers was refined to better align with user preferences by more relevant and specific imagery. The findings emphasize the importance of continuous refinement of AR applications to enhance their educational impact and accessibility.

Center conditions for a cubic differential system with one invariant straight line and one invariant conic

2025-02-17T21:08:50+00:00

In this work we find the center conditions for a cubic system of differential equations with a critical point of a center or a focus type having one invariant straight line and one invariant conic. The center-focus problem is studied by using the Darboux integrability and the rational reversibility methods.