We use an infinite-dimensional Lotka–Volterra model to analyze production, accumulation, and redistribution of wealth in an economy. We show that, if the amount of wealth produced in the economy is small relative to the amount redistributed, the eventual distribution of wealth will be extremely unequal, with all of it being concentrated in single hands in the limit case. The winner’s identity is determined by his ability to redistribute and produce wealth. Similar outcomes are observed in some physical processes. Article is published in the authors’ wording.

The article considers asymptotic distribution of characteristic constants in periodic and antiperiodic boundary-value problems for the second-order linear equation with periodic coefficients. It allows getting asymptotics of stability and instability zones of solutions. It was shown that in the absence of turning points (\(r(t) > 0\)) the instability zones lengths converge to zero with their number increasing, and the stability zones lengths converge to a positive quantity. The situation, when (\(r(t) \geqslant 0\)) and there are zeroes \(r(t),\) results in the fact that the lengths of stability and instability zones have a finite nonzero bound at an unbounded increase of the number of the corresponding zone. But if the function \(r(t)\) is alternating, the lengths of all stability zones converge to zero, and the lengths of instability zones converge to some finite quantities. These conclusions allowed to formulate a series of interesting criteria of stability and instability of solutions of the second-order equation with periodic coefficients. The results given are illustrated by a substantial example. The methods of investigation are based on a detailed study of the so-called special standard equations and the consequent reduction of original equations to any particular type of standard equations. Here, asymptotic methods of the theory of singular perturbance, as well as certain properties of a series of special functions are used.

In the present paper, we consider a multidimensional singularly perturbed problem for an elliptic equation referred to as the stationary reaction-diffusion-advection equation in applications. We formulate basic conditions of the existence of solutions with internal transition layers (contrust structures), and we construct an asymptotic approximation of an arbitrary-order accuracy to such solutions. We use a more efficient method for localizing the transition surface, which permits one to develop our approach to a more complicated case of balanced advection and reaction (the so-called critical case). To justify the constructed asymptotics, we use and develop, to this class of problems, an asymptotic method of differential inequalities, which also permits one to prove the Lyapunov stability of such solutions, as stationary solutions of the corresponding parabolic problems.

In the work, we consider the problem of accelerating the iteration process of the numerical solution of boundary-value problems for partial differential equations (PDE) by the method of collocations and least residuals (CLR). To solve this problem, it is proposed to combine simultaneously three techniques of the iteration process acceleration: the preconditioner, the multigrid algorithm, and the correction of the PDE solution at the intermediate iterations in the Krylov subspace. The influence of all three techniques of the iteration acceleration was investigated both individually for each technique and at their combination. Each of the above techniques is shown to make its contribution to the quantitative figure of iteration process speed-up. The algorithm which employs the Krylov subspaces makes the most significant contribution. The joint simultaneous application of all three techniques for accelerating the iterative solution of specific boundary-value problems enabled a reduction of the CPU time of their solution on computer by a factor of up to 230 in comparison with the case when no acceleration techniques were applied. A two-parameter preconditioner was investigated. It is proposed to find the optimal values of its parameters by the numerical solution of a computationally inexpensive problem of minimizing the condition number of the system of linear algebraic equations. The problem is solved by the CLR method and it is modified by the preconditioner. It is shown that it is sufficient to restrict oneself in the multigrid version of the CLR method only to a simple solution prolongation operation on the multigrid complex to reduce substantially the CPU time of the boundary-value problem solution.

Numerous computational examples are presented, which demonstrate the efficiency of the approaches proposed for accelerating the iterative processes of the numerical solution of the boundary-value problems for the two-dimensional Navier–Stokes equations. It is pointed out that the proposed combination of the techniques for accelerating the iteration processes may be also implemented within the framework of other numerical techniques for the solution of PDEs.

We construct the asymptotics for solutions of a harmonic oscillator with integral perturbation when the independent variable tends to infinity. The specific feature of the considered integral perturbation is an oscillatory decreasing character of its kernel. We assume that the integral kernel is degenerate. This makes it possible to reduce the initial integro-differential equation to an ordinary differential system. To get the asymptotic formulas for the fundamental solutions of the obtained ordinary differential system, we use a special method proposed for the asymptotic integration of linear dynamical systems with oscillatory decreasing coefficients. By the use of the special transformations we reduce the ordinary differential system to the so called L-diagonal form. We then apply the classical Levinson’s theorem to construct the asymptotics for the fundamental matrix of the L-diagonal system. The obtained asymptotic formulas allow us to reveal the resonant frequencies, i. e., frequencies of the oscillatory component of the kernel that give rise to unbounded oscillations in the initial integro-differential equation. It appears that these frequencies differ slightly from the resonant frequencies that occur in the adiabatic oscillator with the sinusoidal component of the time-decreasing perturbation.

We consider a mathematical model of synaptic interaction between two pulse neuron elements. Each of the neurons is modeled by a singularly-perturbed difference-differential equation with delay. Coupling is assumed to be at the threshold, and time delay is taken into consideration. Problems of existence and stability of relaxation periodic movements for obtained systems are considered. It turns out that the ratio between the delay due to internal causes in a single neuron model and the delay in the coupling link between oscillators is crucial.
Existence and stability of a uniform cycle of the problem is proved for the case where the delay in the link is less than a period of a single oscillator that depends on the internal delay. As the delay grows, the in-phase regime becomes more complex, particularly, it is shown that by choosing a suitable delay, we can obtain more complex relaxation oscillation and inside a period interval the system can exhibit not one but several high-amplitude splashes. This means that bursting-effect can appear in a system of two synaptic coupled oscillators of neuron type due to a delay in a coupling link.

Let \(n\in {\mathbb N}\) and \(Q_n=[0,1]^n\). For a nondegenerate simplex \(S\subset {\mathbb R}^n\), by \(\sigma S\) we denote the homothetic copy of~\(S\) with center of homothety in the center of gravity of \(S\) and ratio of~homothety \(\sigma\). By \(\xi(S)\) we mean the minimal \(\sigma>0\) such that \(Q_n\subset \sigma S\). By \(\alpha(S)\) denote the minimal \(\sigma>0\) such that \(Q_n\) is~contained in a translate of~\(\sigma S\). By \(d_i(S)\) we denote the \(i\)th axial diameter of \(S\), i.\,e. the maximum length of~the segment contained in \(S\) and parallel to the \(i\)th coordinate axis. Formulae for~\(\xi(S)\), \(\alpha(S)\), \(d_i(S)\) were proved earlier by the first author. Define \(\xi_n=\min\{ \xi(S): S\subset Q_n\}. \) We always have \(\xi_n\geq n.\) We discuss some conjectures formulated in the previous papers. One of~these conjectures is the following. For~every \(n\), there exists \(\gamma>0\), not depending on \(S\subset Q_n\), such that an~inequality \(\xi(S)-\alpha(S)\leq \gamma (\xi(S)-\xi_n)\) holds. Denote by \(\varkappa_n\) the minimal \(\gamma\) with such a~property. We prove that \(\varkappa_1=\frac{1}{2}\); for \(n>1\), we obtain \(\varkappa_n\geq 1\). If \(n>1\) and \(\xi_n=n,\) then \(\varkappa_n=1\). The equality \(\xi_n=n\) holds if \(n+1\) is an Hadamard number, i.\,e. there exists an Hadamard matrix of~order \(n+1\). This proposition is known; we give one more proof with the direct use of Hadamard matrices. We prove that \(\xi_5=5\). Therefore, there exists \(n\) such that \(n+1\) is not an Hadamard number and nevertheless \(\xi_n=n\). The~minimal \(n\) with such a property is equal to \(5\). This involves \(\varkappa_5=1\) and also disproves the following previous conjecture of the first author concerning the characterization of Hadamard numbers in terms of~homothety of simplices: \(n+1\) is an Hadamard number if and only if \(\xi_n=n.\) This statement is valid only in one direction. There exists a simplex \(S\subset Q_5\) such that the boundary of the simplex \(5S\) contains all the vertices of the cube \(Q_5\). We describe a one-parameter family of simplices contained in \(Q_5\) with the property \(\alpha(S)=\xi(S)=5.\) These simplices were found with the use of numerical and symbolic computations. %Numerical experiments allow to discover Another new result is an inequality \(\xi_6\ <6.0166\). %Прежняя оценка имела вид \(6\leq \xi_6\leq 6.6\). We also systematize some of our estimates of numbers \(\xi_n\), \(\theta_n\), \(\varkappa_n\) derived by~now. The symbol \(\theta_n\) denotes the minimal norm of interpolation projection on the space of linear functions of \(n\) variables as~an~operator from \(C(Q_n)\) to~\(C(Q_n)\).

When investigating piecewise polynomial approximations in spaces \(L_p, \; 0~<~p~<~1,\) the author considered the spreading of k-th derivative (of the operator) from Sobolev spaces \(W_1 ^ k\) on spaces that are, in a sense, their successors with a low index less than one. In this article, we continue the study of the properties acquired by the differentiation operator \(\Lambda\) with spreading beyond the space \(W_1^1\)
$$
\Lambda~:~W_1^1~\mapsto~L_1,\; \Lambda f = f^{\;'}
$$.
The study is conducted by introducing the family of spaces \(Y_p^1, \; 0 <p < 1,\) which have analogy with the family \(W_p^1, \; 1 \le p <\infty.\) This approach gives a new perspective for the properties of the derivative. It has been shown, for example, the additivity property relative to the interval of the spreading differentiation operator:
$$ \bigcup_{n=1}^{m} \Lambda (f_n) = \Lambda (\bigcup_{n=1}^{m} f_n).$$
Here, for a function \(f_n\) defined on \([x_{n-1}; x_n], \; a~=~x_0 < x_1 < \cdots <x_m~=~b\), \(\Lambda (f_n)\) was defined. One of the most important characteristics of a linear operator is the composition of the kernel.
During the spreading of the differentiation operator from the space \( C ^ 1 \) on the space \( W_p ^ 1 \) the kernel does not change. In the article, it is constructively shown that jump functions and singular functions \(f\) belong to all spaces \( Y_p ^ 1 \) and \(\Lambda f = 0.\) Consequently, the space of the functions of the bounded variation \(H_1 ^ 1 \) is contained in each \( Y_p ^ 1 ,\) and the differentiation operator on \(H_1^1\) satisfies the relation \(\Lambda f = f^{\; '}.\)
Also, we come to the conclusion that every function from the added part of the kernel can be logically named singular.