The article shows an experimental (part 2) improvement in the stabilization of the gaze under galvanic vestibular stimulation.

We investigate the statement of the optimization inverse spectral problem with incomplete spectral data for the one-dimensional Schr¨odinger operator on the entire axis: for a given potential ????0, find the closest function ????^ such that the first ???? eigenvalues of the Schrodinger operator with potential ????^ coincided with the given values ????* ∈ R, ???? = 1, ????.

This paper is devoted to orthorecursive expansions introduced in 1999, which generalize the notion of orthogonal expansions. We present main results obtained over the last 20 years concerning properties of orthorecursive expansions and indicate some areas for further research.

The article shows a theoretical (part 1) improvement in thestabilization of the gaze in galvanic vestibular stimulation.

We consider the following optimization inverse spectral problem: for a given matrix potential Q_0(x) , find a matrix function Q̂(x) closest to it and such that the first eigenvalue of the Sturm–Liouville matrix operator with the potential Q̂(x) coincides with a given value λ _1^*∈ℝ . The main result of the paper is to establish a new type of connection between the specified inverse spectral problem and systems of second-order nonlinear differential equations known as systems of nonlinear Schrödinger equations in mathematical physics.

The global demographic transition and phenomenological models of demographic dynamics by S.P. Kapitsa and B.M. Dolgonosov are considered, which are based on the principles of the demographic and information imperatives. The technological imperative is analyzed, which is intermediate between the former two. An original mathematical model to predict demographic dynamics in the era of widespread use of intelligent machines is proposed. The model was built using the demographic model of Kapitsa and a formula to produce useful information in human society, based on the assumption by S. Kuznets about technological progress. The calculations made using this model show that the population of the Earth, having reached a maximum value of 8.37 billion people in 2050, will then begin to decline steadily and by 2100 will not exceed 7.9 billion people.

By using the US economy as an example, the paper shows how the COVID-19 pandemic has changed its short-term dynamics, causing a deep crisis recession in 2020 rather than the expected short-term and shallow recession in 2022 caused by the inflation of the financial bubble during the credit expansion that followed the financial and economic crisis of 2008–2009. To predict the latter scenario, which is natural for the US economy, the authors first developed a mathematical model based on Hyman Minsky’s theory of financial instability, which can serve to manage the processes of credit expansion and contraction in an unstable economy.

Асташова И.В., Барабанов Е.А., Боровских А.В., Бутузов В.Ф., Быков В.В., Ветохин А.Н., Глызин С.Д., Горицкий А.Ю., Денисова Н.В., Изобов Н.А., Ильин А.В., Ильяшенко Ю.С., Капустина Т.О., Кигурадзе И.Т., Козлов В.В., Колесов А.Ю., Коньков А.А., Ломов И.С., Моисеев Е.И., Палин В.В. и др.//Дифференциальные уравнения, 2021

For the differential equation $$y^{\prime \prime \prime }(x)=\lambda y(x)$$ on the interval $$[0,1]$$ , we consider the three-point eigenvalue problem $$a_iy^{(i-1)}(0)+y(c)+b_i y^{(i-1)}(1)=0$$ , $$i=1,2,3$$ , where $$\lambda$$ is the spectral parameter, the point $$c\in (0,1)$$ is fixed, and $$a_i$$ and $$b_i$$ , $$i=1,2,3$$ , are somecomplex numbers. Necessary and sufficient conditions that the coefficients $$a_i$$ and $$b_i$$ must satisfy for the indicated three-point problemto have degenerate boundary conditions are obtained.

A mathematical model of the formation of output information in a biosensor of angular acceleration is presented. The functional and numerical parameters of the model have been determined by results of experiments made in 2001–2008. A comparison with the mathematical model of J. M. Goldberg and C. Fernandez (1971) describing the change in spike frequency of the primary afferent neuron spikes in response to an angular acceleration of the head as it turns around a vertical axis is carried out.

The well-known mathematician in the theory of functions of a real variable and a leading expert in mathematical education Mikhail Konstantinovich Potapov observed his 90th birthday on 29 January 2021. Potapov was born in Pyatigorsk and graduated from Pyatigorsk Pedagogical Institute in 1952 as a teacher of mathematics and physics in secondary school. Subsequently, after he developed into a prominent figure in mathematics, he remained always mindful of the teaching of mathematics in school and he wrote innovative textbooks. He completed his postgraduate studies in the Faculty of Mechanics and Mathematics at Moscow State University (MSU), with S. M. Nikol’skii as his scientific advisor. Since then Potapov’s research and teaching activities have been connected with MSU, where he has been one of the leading professors in the Faculty of Mechanics and Mathematics for decades. He is the author of more than 250 research papers, and the total number of his publications exceeds 800. The main topics of his investigations are the theory of approximations of functions, embedding theorems, and trigonometric series. He was one of the first authors to study approximations of functions by algebraic polynomials in an integral metric. In the 1950s he proved Jackson’s theorem for Lipschitz classes in the spaces Lp, 1 ⩽ p < ∞. He described various structural characteristics of classes of continuous functions on a closed interval or a half-line that have one or another order of best approximation by algebraic polynomials, and he answered the question of the stability of these characteristics in the classical cases of Jacobi and Laguerre weights. He proved Jackson’s theorem and its converse for best approximation by algebraic polynomials and the moduli of smoothness defined in terms of symmetric

The article provides evidence that most of the cognitive work in the digital age will continue to be reserved for human labor, since this kind of work can generally be fragmented into nonprogrammable tasks (50–75%), the solution of which requires human creative work, and routine programmable tasks that can be solved by intelligent machines (IMs). The authors propose a mathematical model for calculating labor productivity in the digital economy, characterized by widespread “human+IM” symbiosis. The calculations performed on the proposed model demonstrate that: 1) the human+IM symbiosis uses digital technologies to realize potential opportunities of increasing labor productivity in the economy; 2) the highest level of labor productivity is achieved if human labor prevails in the human+IM symbiosis, while the lowest level of labor productivity is observed if the share of programmable IM-performed work prevails; 3) in developed countries labor productivity of 3% per year can be achieved by the mid-2020s, and this level can be retained until the 2040s.

We study boundary conditions for the diffusion operator defined on a star-shaped geometric graph consisting of three edges with a common vertex. We show that if the edge lengths are pairwise distinct, then there do not exist degenerate boundary conditions for the diffusion operator. If the edge lengths coincide and the potentials are symmetric, then the characteristic determinant of a boundary value problem for the diffusion operator cannot be a constant other than zero, and the set of boundary value problems for which the characteristic determinant is identically zero is infinite (a continuum). We show that, for the diffusion operator on the star-shaped graph, the set of boundary value problems whose spectrum fills the entire plane consists of eighteen classes, each of which contains eight to nine arbitrary constants. Recall that for the diffusion operator defined on an interval this set consists of two problems.

Доказана теорема единственности восстановления дифференциального оператора n-го порядка с нераспадающимися краевыми условиями по нескольким спектрам. Эта теорема опирается на результаты Е.А. Барановой.

Abstract We consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

A uniqueness theorem for reconstruction of an nth-order differential operator with nonseparated boundary conditions from several spectra is proved. This theorem is based on the results of E.A. Baranova.

We study the inverse spectral problem of reconstructing nonsplitting boundary conditions for the Sturm–Liouville operator from a minimal amount of spectral data. Necessary and sufficient conditions are derived for the existence of solutions of the problem under consideration and for these solutions to be isolated or nonisolated. We propose a simple analytical algorithm for reconstructing boundary conditions. (Explicit formulas are provided for the coefficients of the boundary conditions.) It is shown that the condition determining the properties of the problem is the dimension of the linear span of matrices of fundamental solution systems corresponding to the given spectral data.