Gur'eva V.M., Morohotova L.S., Kotov Yu.B., Semenova T.A.
Mathematical methods for processing results of the study of interaction in complex systems have been developed. Logical symptoms have been used to treat biochemical tests of patients. This work is a part of the classification problem for patients and prediction of severe complications during pregnancy with type-2 diabetes. The technology involving masks, as well as nonparametric statistics, has been used. Medical information has been obtained for 95 pregnant women in different gestational periods. The data array with a complex structure has been investigated in two stages. In the first stage, a list of logical symptom vectors has been created. Using masks and indistinguishability of vectors from them, data suitable for comparison have been selected. As a result, the space of medical symptoms has been reduced. The set of variables has been divided into six different groups. In the second stage, the formal intergroup mask has been built and new indistinguishability values related to the mask have been calculated. The separation rule obtained has a high significance level of 0.001 (Fischer probability pF = 2.1 × 10–25) between a class of patients with severe obstetric complications and other classes. As a result, nine medical rules have been obtained for evaluating medical conditions for patients, manifested in the first 26 weeks of pregnancy. The rules became interesting for physicians. Keywords: classification of patients, method of logical symptoms, mask, Fisher’s exact method, Smirnov two-sample criterion, indistinguishability, gestosis, biochemical tests
Maslennikov S.P., Serebryakova A.S., Komarov D.A., Maslennikova I.S.
High-voltage solid-state switches with a modular structure are developed on the basis of series connected MOSFETs. The switches make it possible to increase the working voltage up to several tens of kilovolts by means of their scaling and can be used in generators of high-voltage microsecond pulses for various applications. Switch control circuits based on pulsed transformers are suggested and tested, which make it possible to widely adjust on-state duration of the switches and to create conditions for their stable switching with nanosecond response. The transistor modules of the compound switches are based on the electrical circuits with unipolar or bipolar control modes of the power transistors. The results of tests carried out for high-resistive loads show that the commutation stability of the switches is ensured for transistor module circuits with bipolar control. The switches with operating voltages up to 10 kV and pulsed currents up to 12 A are experimentally implemented. A single-stroke pulse generator has been developed on the basis of a compound switch with a bipolar control mode at the output stage. The tests of the generator have shown the stability of its operation in the pulse duration range from 1 μs to 0.5 ms at amplitude voltages up to 10 kV and load resistances in the range of 103–107 Ω. Keywords: semiconductor switch, high-voltage pulse generator, MOSFET
Tyuflin S.A., Nagornov O.V.
Information about the past climate of the Earth is contained in various indirect sources. One of such sources is the measured temperature-depth profile in glaciers because temperature signals from the surface penetrate into a glacier and disturb the steady-state temperature profile in it. The reconstruction of the surface temperature requires the solution of the inverse problem for the thermal diffusivity equation with vertical advection of annual layers. The measured temperature-depth profile is the so-called redetermination condition. The properties of solution for this problem have been studied. It has been proven that the solution is not unique. It has been demonstrated that the effects of several significantly different surface signals to the glacier thickness cannot be distinguished within the accuracy of measurements. This means that numerous paleotemperature reconstructions based on the borehole temperature data have to be reconsidered. Keywords: inverse problems, paleoclimate, past surface temperature reconstruction
Lavrova S.F., Kudryashov N.A., Sinelshchikov D.I.
The standard perturbed FitzHugh–Nagumo model that describes impulse propagation between neurons is considered. This model is a simplification of the Hodgkin–Huxley model, which is inconvenient because involves a large number of variables and experimentally obtained parameters. The standard FitzHugh–Nagumo model is described by a system of two differential equations, contains three parameters, does not pass the Painlevé test, and does not have any physically interesting analytical solutions. That is why the perturbation of this model by adding a cubic term to one of the equations of the system is studied. The exact kink solution of the perturbed FitzHugh–Nagumo model, which can be used as a transfer function in construction of neuron nets, has been obtained. The stability of the stationary points of the system has been studied. The numerical simulation of the studied model has been performed. It has been shown that the numerical solution of the perturbed system with a small perturbation parameter is close to the corresponding solution of the standard model. The effect of the parameters on the dynamics of solutions of the perturbed model has been analyzed. Keywords: neuron, Hodgkin–Huxley model, FitzHugh–Nagumo model, nonlinear differential equations, stationary points, dependence on the parameters, numerical solution
Mayorov V.V., Mayorova N.L., Kaschenko S.A.
In classical neural networks, information is reflected as a set of stable equilibrium states. A cycle of works concerning the development of neural network models without equilibrium states has been reviewed. Oscillating regimes in neural networks are interpreted as images reflecting some facts. Based on physiological perceptions, a model of neural environment is proposed to store information in a waveform. Effective analytical methods for the asymptotic analysis of nonlinear nonlocal oscillations in the system of equations with delay describing a neural population are discussed. Theoretical studies made it possible to effectively solve particular important problems. On the basis of the delay equation, a new model of a neuron generating short-time, high-amplitude pulses has been developed and investigated. It has been shown that the specific form of the functions fNa and fK entering the equation is not significant under certain very natural assumptions. A model of the interaction between neurons and, thus, a neural network has been proposed. New mathematical methods for studying systems of equations for a neural population have been developed. They have been used to analytically prove that the annular system of model neurons can store predetermined periodic sequences of pulses (can store information). It is proved that the adaptation of the neuron interaction force in the model forms a population generating a given periodic sequence of pulses (the ability to record information). A model of synchronization of neural structures is constructed, which makes it possible to establish the identity of pulse sequences. It is shown that sequences of pulses can be stored for a short time in the ring population of neural units without adjustment of weights, (a model of short-term wave memory). Keywords: neural networks, equilibrium states, oscillation regimes, asymptotic analysis, equations with delay
The analytical properties of solutions for one class (including four families) of three-dimensional dynamical dissipative systems with quadratic nonlinearities have been investigated. The absence of a chaotic behavior is a common qualitative property of all systems of this class. Under the assumption that the independent variable is complex, the Painlevé analysis of solutions for each of the systems of this class has been performed. The nature of the movable singularities for solutions of the systems has been studied within two approaches. The first approach is based on the reduction of some systems to equivalent differential equations of the second or third order with a polynomial (with respect to the unknown function and its derivatives) right-hand side and subsequent comparison of the obtained equations with the well-known Painlevé-type equations. The second approach is based on the application of the Painlevé test to the other systems of this class. It has been found that none of the systems (with one exception) is a Painlevé-type system. Moreover, these systems include systems in which one component of the solution has no movable singular points at all. The general solution of one system, which is a Painlevé-type system, has also no movable singular points. Autonomous third-order equations that neither belong to the Painlevé type nor show a chaotic behavior have been separated. Keywords: dynamical systems, Painlevé test, Painlevé property, movable singular point
Polyanin A.D., Shingareva I.K.
Hypersingular nonlinear boundary-value problems with a small parameter Ε at the highest derivative are described. These problems essentially (qualitatively and quantitatively) differ from the usual singularly perturbed boundary-value problems and have the following unusual properties. (i) In hypersingular boundary-value problems, hyperfine boundary layers arise, and the derivative at the boundary layer can have a very large value of the order of e1/ε (in typical problems with boundary layers, the derivative at the boundary has the order of ε–1 or less). (ii) In hypersingular boundary-value problems, the position of the boundary layer is determined by the specified boundary values (in typical problems with boundary layers, the position of the boundary layer is determined by the coefficients of the given equation and is independent of the magnitude and sign of the boundary values). (iii) Hypersingular boundary-value problems do not admit the direct application of the method of matched asymptotic expansions (without a preliminary nonlinear point transformation of the equation under consideration). Examples are given and exact solutions are obtained for hypersingular boundary-value problems with ordinary differential equations and equations in partial derivatives. It is shown that for some boundary values there can be no boundary layers as ε → 0. Keywords: hypersingular boundary-value problems, differential equations with a small parameter, nonlinear boundary-value problems, boundary layers, linearization and exact solutions
In 1965, Martin Kruskal and Norman Zabusky introduced the notion of soliton for an extraordinary solitary wave appearing in nature, physics, and other sciences. In 1967, the inverse scattering transform method was developed to solve the Cauchy problem for the Korteweg–de Vries equation. These events were outset for soliton theory. In view of these remarkable data, the history of development of soliton theory is reviewed. The discovery of a solitary wave on the Edinburgh–Glasgow channel by Jhon Scott Russel in 1834 is described. It is discussed how Korteweg and de Vries derived an equation now called the Korteweg–de Vries equation whose solution describes the wave observed by Russel. One of the first numerical experiments performed in Los Alamos for a one-dimensional anharmonic lattice, which led to Fermi–Pasta–Ulam paradox, is described. It is discussed how Martin Kruskal and Norman Zabusky derived the Korteweg–de Vries equation from the Fermi–Pasta–Ulam mathematical model and explained the Fermi–Pasta–Ulam paradox by the properties of the soliton. Applications of solitons to describe tsunami, Rossby waves, dislocations in solids, and superconductivity, as well as the application of optical solitons in optical information transmission lines, are discussed. Keywords: mathematical model, soliton, Korteweg–de Vries equation, nonlinear Schrödinger equation, Kadomtsev–Petviashvili equation, sine-Gordon equation, Fermi–Pasta–Ulam paradox, Cauchy problem, inverse scattering transform
Kashchenko I.S., Kaschenko S.A.
The local dynamics of two-component systems of nonlinear parabolic equations, where one of the diffusion coefficients is sufficiently small, has been studied. Both cases where nonlinearity depends only on an unknown function and cases of strong nonlinearity, which also depends on its derivative, have been analyzed. It has been shown that the critical cases in the problem of stability of equilibrium can have infinite dimension. Various relations between two small parameters of the problem—supercriticality and diffusion coefficient—are studied. In critical cases, special families of nonlinear parabolic equations that do not contain small parameters are constructed. These systems play the role of normal forms: their nonlocal dynamics mainly determines the behavior of the solutions of the original system of equations. It has been shown that these equations are reduced to families of parabolic boundary-value problems. Explicit formulas for their coefficients and formulas connecting the solutions of analogues of normal forms and the original problem have been obtained. It has been shown that an arbitrarily large number of periodic solutions can be formed near the equilibrium state because of bifurcation, and they are usually formed on modes with asymptotically large numbers. Critical cases with additional degeneracy when the codimension of the resulting bifurcation is 2 have been studied separately. Keywords: parabolic system, singular perturbation, dynamics, normal form, contrast system