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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.

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

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.

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.

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.

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.

Homo sapiens and naked mole rats (Heterocephalus glaber) are vivid examples of social mammals that differ from their relatives in particular by an increased lifespan and a large number of neotenic features. An important fact for biogerontology is that the mortality rate of H. glaber (a maximal lifespan of more than 32 years, which is very large for such a small rodent) negligibly grows with age. The same is true for modern people in developed countries below the age of 60. It is important that the juvenilization of traits that separate humans from chimpanzees evolved over thousands of generations and millions of years. Rapid advances in technology resulted in a sharp increase in the life expectancy of human beings during the past 100 years. Currently, the human life expectancy has exceeded 80 years in developed countries. It cannot be excluded that the potential for increasing life expectancy by an improvement in living conditions will be exhausted after a certain period of time. New types of geroprotectors should be developed that protect not only from chronic phenoptosis gradual poisoning of the body with reactive oxygen species (ROS) but also from acute phenoptosis, where strong increase in the level of ROS immediately kills an already aged individual. Geroprotectors might be another anti-aging strategy along with neoteny (a natural physiological phenomenon) and technical progress.

In neoclassical economic theory, the dynamics of development is mainly described by the Solow production function (Solow model), which determines the growth of the aggregate income of a country or an individual company depending on the main economic factors, such as physical capital, labor, and technological progress, driven by technological revolutions. In the upcoming digital economy, the key role will be played by technological information, which includes the knowledge and know-how needed to create high-quality goods and services that meet individual preferences, rather than mass demand, as before. It follows that the most appropriate mathematical model for describing the economic dynamics in the digital age is the half-forgotten Ramsey–Cass–Koopmans model with optimization of the consumer behavior of a representative household. This paper shows that this model implies the need to ensure the exponential growth of technological information in order for the economy to move along an optimum development trajectory. The main advantage of the Ramsey–Cass–Koopmans model is the transformation of practically significant economic development tasks into the problem of optimizing the corresponding Lagrangian or Hamiltonian, with each Lagrangian (Hamiltonian) generating a certain mode of production of technological information. Thus, it is through the Lagrangian (Hamiltonian) that the functional connection between economic growth and the dynamics of the production of technological information is established. For the first time, a formula has been obtained that links the pace of technological progress with the pace of production of technological information and can be the basis for developing an information model of economic dynamics. Examples of real modes of production of technological information used to calculate the trajectories of technological progress in the information–digital era are given.

We study the boundary conditions of the Sturm-Liouville problem posed on a star-shaped geometric graph consisting of three edges with a common vertex. We show that the Sturm-Liouville problem has no degenerate boundary conditions in the case of pairwise distinct edge lengths. However, if the edge lengths coincide and all potentials are the same, then the characteristic determinant of the Sturm-Liouville problem cannot be a nonzero constant and the set of Sturm-Liouville problems whose characteristic determinant is identically zero and whose spectrum accordingly coincides with the entire plane is infinite (a continuum). It is shown that, for one special case of the boundary conditions, this set consists of eighteen classes, each having from two to four arbitrary constants, rather than of two problems as in the case of the Sturm-Liouville problem on an interval.

Рассматривается система двух обобщенных уравнений Хатчинсона, связанных посредством линейных диффузионных слагаемых. Устанавливается, что при подходящем выборе параметров в упомянутой системе существует устойчивый релаксационный цикл, компоненты которого переходят друг в друга при некотором фазовом сдвиге. Приводится также ряд дополнительных свойств этого цикла, позволяющих интерпретировать его как режим самоорганизации.

The boundary conditions of the Sturm–Liouville problem defined on a star-shaped geometric graph of three edges are studied. It is shown that, if the lengths of the edges are different, then the Sturm–Liouville problem does not have degenerate boundary conditions. If the lengths of the edges and the potentials are identical, then the characteristic determinant of the Sturm–Liouville problem cannot be equal to a constant different from zero. However, the set of Sturm–Liouville problems for which the characteristic determinant is identically zero is infinite (continuum). In this way, in contrast to the Sturm–Liouville problem defined on an interval, the set of boundary value problems on a star-shaped graph whose spectrum completely fills the entire plane is much richer. In the particular case when the minor $${{A}_{{124}}}$$ of the coefficient matrix is nonzero, this set consists of not two problems, as in the case of the Sturm–Liouville problem given on an interval, but rather of 18 classes, each containing two to four arbitrary constants.

The inverse Sturm-Liouville problem with nonseparated boundary conditions on a star-shaped geometric graph consisting of three edges with a common vertex is studied. It is shown that the Sturm-Liouville problem with general boundary conditions cannot be reconstructed uniquely from four spectra. A class of nonseparated boundary conditions is obtained for which two uniqueness theorems for the solution of the inverse Sturm-Liouville problem are proved. In the first theorem, the data used to reconstruct the Sturm-Liouville problem are the spectrum of the boundary value problem itself and the spectra of three auxiliary problems with separated boundary conditions. In the second theorem, instead of the spectrum of the problem itself, one only deals with five of its eigenvalues. It is shown that the Sturm-Liouville problem with these nonseparated boundary conditions can be reconstructed uniquely if three spectra of auxiliary problems and five eigenvalues of the problem itself are used as the reconstruction data. Examples of unique reconstruction of potentials and boundary conditions of the Sturm-Liouville problem posed on the graph under study are given.

A system of two generalized Hutchinson’s equations coupled by linear diffusion terms is considered. It is established that for an appropriate choice of parameters, the system has a stable relaxation cycle whose components turn into each other under a certain phase shift. A number of additional properties of this cycle are presented that allow one to interpret it as a self-organization mode.

Nonsingular endomorphisms of the m-torus $$\mathbb{T}$$ m, m ≥ 2, which are C1 perturbations of linear hyperbolic endomorphisms are considered. Sufficient conditions for such maps to be hyperbolic (i.e., belong to the class of Anosov endomorphisms) are found.

The application of galvanic vestibular stimulation is considered for the galvanic correction of pilot’s vestibular activity during visual flight control on a flight-dynamic stand and in extreme flight situations.

Сам жанр журнала "Успехи математических наук" предполагает обзор прежде все-го математических результатов Владимира Андреевича Успенского. Однако вся его личность и деятельность обязывают к более широкому взгляду на вещи. Владимир Андреевич был выдающимся математиком, и это проявлялось в различных сферах его деятельности и её результатах и вне математики. Об этих сферах также будет идти речь в нашем обзоре. При жизни В. А. Успенский обходился без юбилейных статей в "Успехах". Это делает особенно важным опубликовать, хотя и с опозданием, его биографию как учёного в УМН.

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.

Uniqueness theorems are proved for two inverse problems for a fourth-order differential operator with nonseparated boundary conditions. The first of the problems, which has technical applications, is the problem of identification of a differential equation and two boundary conditions, and the second problem is the problem of identification of a differential equation and four boundary conditions. One of two data sets is used as the spectral data of the problem. The first data set is the spectrum of the problem itself (or three of its eigenvalues) and the spectral data of a system of three problems, and the second data set is the spectrum of the problem itself (or three of its eigenvalues) and the spectra of ten boundary value problems.

Mathematical models for practical calculations of technological progress (total productivity of production factors) and economic growth in the era of widespread digitalization and robotization of national economies, where the main factor of production is technological information, are developed and verified. For this purpose, models using different modes of information production are proposed for the first time. It is shown that the economic effect of the digitalization of an economy will not come immediately, but with a lag of about eight years. For the US economy, forecast calculations show that this will happen in 2022–2026 with total productivity increasing by 1.1 percentage points up to 2.5% per year.

We consider mappings of the m-dimensional torus Tm (m ≥ 2) that are C 1-perturbations of linear hyperbolic automorphisms. We obtain sufficient conditions for such mappings to be one-to-one hyperbolic mappings (i.e., Anosov diffeomorphisms). These results are used to study the blue-sky catastrophe related to the vanishing of a saddle-node invariant torus with a quasiperiodic winding in a system of ordinary differential equations.

Theorems on the unique reconstruction of a Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions are proved. Two spectra and finitely many eigenvalues (one spectrum and finitely many eigenvalues for a symmetric potential) of the problem itself are used as the spectral data. The results generalize the Levinson uniqueness theorem to the case of nonsplitting boundary conditions containing polynomials in the spectral parameter. Algorithms and examples of solving relevant inverse problems are also presented.

We consider circular chains of unidirectionally coupled ordinary differential equations which are mathematical models of artificial gene networks. We study the problems of the existence and stability of special periodic solutions, the so-called travelling waves, in these chains. We establish that the number of such periodic solutions grows unboundedly as the number of links in the chain grows. However, at most one of these travelling waves can be stable. We give an explicit algorithm for choosing the stable cycle.

We prove theorems on the solvability of the inverse Sturm-Liouville problem with nonseparated conditions by two spectra and one eigenvalue and theorems on the unique solvability by two spectra and three eigenvalues. We find exact and approximate solutions of the inverse problems. Related examples and counterexample are given.

In this paper, we obtain a representation for the regularized traces (1) of evenorder differential opera� tors in terms of Hamiltonian functional integrals and the corresponding Feynman formulas. We also present an example showing how such a representation can be applied to obtain formulas not containing functional integrals. A Feynman formula (2) is a representation of a solution of an evolution differential equation or an object somehow related to this equation in terms of limit integrals over Cartesian powers of some space. If the space under consideration coincides with the spa� tial domain of the solution, then the corresponding formulas are said to be Lagrangian; if this is the prod� uct of this spatial domain and the dual of the linear space containing it, then the formulas are said to be Hamiltonian. A similar terminology is used for func� tional integrals being approximated by Feynman for� mulas. In this paper, we extend results of (3) to higher� order differential operators, which requires substan� tially modifying the technique; namely, we use Hamil� tonian (rather that Lagrangian, as in (3)) Feynman formulas and functional integrals. Moreover, in those Hamiltonian functional integrals which are used in the paper, integration is over the set of functions taking values in the product of the momentum space and a domain in the configuration space not coinciding with the entire space. Main attention is paid to the algebraic aspects of the problems under consideration; some analytic assumptions are omitted.