In the paper, we estimate the cardinality of the set of positive integers 1 ⩽ 𝑏1, 𝑏2, . . . , 𝑏𝑠 ⩽ 𝑋 satisfying to the conditions of congruent-solvability and positive solvability, and of the exceptional set in the problem of simultaneous representation of 𝑠 numbers 𝑏1, 𝑏2, . . . , 𝑏𝑠 the sums of 𝑚(𝑠 < 𝑚 < 2𝑠) primes. We also obtain a new lower estimate for the number of such representations.

The Gromov–Hausdorff distance (hereinafter referred to as the GH-distance) is a measure of non-isometricity of metric spaces. In this paper, we study a modification of this distance that also takes topological differences into account. The resulting function of pairs of metric spaces is called the continuous GH-distance. We show that many basic properties of the classical GH-distance also hold in the continuous case. However, the continuous GH-distance, distinguishing between topologies, can differ significantly from the classical one. We will provide numerous examples of this distinction and demonstrate the role of topological dimension here.
In particular, we will prove that the continuous GH-distance, like the classical one, is intrinsic, but, unlike the classical one, it is incomplete. Since we are dealing with all metric spaces, we will show, within the framework of the von Neumann-Bernays-G¨odel theory, how topological concepts can be transferred to proper classes.

Modern mathematical models, computer technologies, financial instruments and mechanisms have formed a new scientific sphere – "financial engineering". In the context of financial engineering, the formulation of new mathematical problems of financial resource management, including the modification of target functionals, is of interest. In this paper, one of the variants of such modification is proposed, namely, for a two-sector model of economic dynamics, a twocriteria problem is considered formalized as a maximin control problem. A complete study of the dependence of the type of optimal trajectory on the value of the control interval is carried out.

This paper studies the properties of unimodular lattices of simultaneous Dirichlet approximations and reciprocal lattices of simultaneous Dirichlet approximations. A theorem is proved stating the equality of distances between two lattices and between two corresponding reciprocal lattices. The completeness of the spaces of lattices of simultaneous Dirichlet
approximations and mutual lattices of simultaneous Dirichlet approximations is proved.

In this paper we study the phase topology for the Kovalevskaya – Chaplygin integrable case in rigid body dynamics. On the one hand, it is the generalization of the classical Kovalevskaya and Chaplygin cases. On the other hand, it is inscribed in the 6-parameter family of partially
integrable (under zero value of the area integral) Hamiltonian systems with two degrees of freedom. For the given problem, we study in details the critical subsystems — Hamiltonian systems with one degree of freedom which are restrictions of the initial system to the critical
set of the momentum mapping. We obtain an explicit parametrization of the critical set which gives the bifurcation diagram and the image of the momentum mapping. For all five critical subsystems we provide their explicit solutions in elliptic quadratures under constant value of
the energy integral and the parameter of the problem. Besides that, for each critical subsystem we describe the bifurcations of the integral trajectories under the change of the energy level. It turns out that all non-trivial bifurcations of the saddle type are 2-atoms 𝐵 and 𝐶2 (standard transformations of two critical circles into one or two circles respectively).

This article derives formulas for free angles of various orders of RR-polytopes and applies the resulting relations to prove the completeness of the list of non-composite RR-polytopes of the second type with acute-angled rhombic vertices. Free angles of the first order are flat angles whose vertices belong to the rhombic stars of the RR-polytopes. The sides of each free angle of the first order are two sides of adjacent rhombi of the rhombic star. Previously, the author found a relationship between the acute angles of the rhombic vertex rhombi and free angles of the first order. Here, we will establish relationships between the flat angles between two sides of regular polygons glued into free angles of the first order and the acute angles of the rhombi.
The angles between the sides of regular faces are called free angles of the second order in this article. Similarly, the sides of adjacent regular polygons glued into free angles of the second order form an angle called a free angle of the third order. All possible cases of gluing one or two identical regular polygons into free angles are considered, which makes it possible to establish the completeness of the list of non-composite 𝑅𝑅-polyhedra with acute-angled rhombic vertices and regular faces of various types.

In this paper, we consider the modified Korteweg–de Vries equation with an integral source.
It is shown that the inverse spectral problem method can be applied to integrate the modified Korteweg–de Vries equation with an integral source. The evolution of the spectral data of the Dirac operator with a periodic potential associated with the solution of the modified
Korteweg–de Vries equation with an integral source is determined. The solvability of the Cauchy problem for the infinite system of Dubrovin–Trubowitz differential equations in the class of six times continuously differentiable periodic functions is proved. It is shown that the  constructed solution, indeed, satisfies the equation under consideration.

In this article we show that in a regular 𝑝-space, for every pair of disjoint ideal Rothberger set and closed set there is a pair of disjoint open sets such that one contains the closed set and other one’s complement with respect to the ideal rothberger set is in the corresponding sub ideal. Moreover, we demonstrate how families of closed sets can be used to describe the ideal Rothberger spaces.

In this article we study retract and weakly retract lattices — lattices whose congruences are generated by retractions or weakly retractions, respectively. A retraction (weak retraction) of a lattice is any idempotent lattice (semilattice) endomorphism.
We have obtained the structural properties of retract and weakly retract lattices (section 2).
It is proved that the class of all retract lattices is closed under homomorphic images (Theorem 1), finite direct products (Theorem 2), direct sums (Theorem 4), and passage to dual lattices (Remark 13), but not under taking sublattices (Proposition 1) and ordinal sums (Example 12). Example 11 shows that the finite products of chains are retract lattices. A wider class of weakly retract lattices is closed under homomorphic images, finite direct products, direct sums, and ordinal sums (Theorem 3).
In section 3, preliminary results are presented on the retractions of the direct product of an 𝑚-element and an 𝑛-element chain (Proposition 2, Examples 13 and 14). The problem of finding the number of retractions of such a product is posed.
Section 4 contains the first author’s results on the structure of retract semilattices, which complement the results on retract and weakly retract lattices.
Explanatory notes are made.

This paper logically analyzes a common real-life (or literary) scenario in which all listeners (or, accordingly, readers) suddenly burst into laughter after a certain phrase spoken by the narrator (or author). It turns out that, from a mathematical logic perspective, this occurs when, in a somewhat challenging situation, something is said (or written) that is completely unexpected for the listener (or reader), but to a certain extent justified, although perhaps not logically correct. A couple of detailed examples of humorous situations are provided to
demonstrate the stated assertion, as well as several other similar examples for independent perception and analysis of the concept of humor.

The convergence analysis of the spectral element method (one of the modern modifications of the finite element method) for the dynamic problem of elasticity theory is performed by comparing the numerical solution with the analytical solution of the Lamb problem — the
problem of dynamic action on the boundary of a half-plane or half-space by a concentrated or distributed load changing according to some time law. The article considers the effect on the boundary of a load changing according to the Berlage time law. The calculations are
performed using the domestic strength software package “Fidesys”. Stress distribution graphs for the material under study are given. The dependence of the error of the numerical solution on the order of elements for a fixed number of points per Rayleigh wavelength is investigated.

During the 17-th century, analytical methods were formed in the works of European scientists, replacing geometric and synthetic ones, where for each problem their own unique specific method was created, which did not allow generalization to a wide class of problems.
Based on the generalization of analytical methods, I. Newton and G.W. Leibniz created their theories. Their presentation was difficult to master. Neither Newton nor Leibniz had direct students. In England, С. Maclaurin, E. Halley, A. de Moivre and D. Stirling took on the
propaganda of Newton’s doctrine. In Europe, the Bernoulli brothers took up the dissemination of Leibniz’s doctrine. The period under consideration is a transitional period from the era of classical geometric methods to universal analytical ones. Jacob and Johann Bernoulli were the best teachers of mathematics in Europe; no university gave such a volume of knowledge. Both in Basel and in Paris, they had many students and followers. Thanks to their teaching activities, the Basel mathematical school, the strongest in Europe, was formed. Groups of scientists who studied or consulted with Jacob Bernoulli and Johann Bernoulli, both personally and in correspondence, both regularly and occasionally, are identified, and their scientific activities are characterized. This generation, in turn, created the potential for the next generation and the further development of analytical methods, thanks to the generalization and classification of problems of analysis and analytical mechanics, by the middle of the 18-th century the architecture of mathematics had changed and its areas had expanded.