After introducing some basic mathematical theory on kleinian groups, we discuss both technical features and drawbacks of the classic approach adopted to render the digital representation of diagrams that display either the action of tessellation and the limit sets related to Kleinian groups in one complex variable. We will then follow to illustrating the details of a new approach, based upon numerical base conversion. This discussion will be supported by pseudo-code, in order to explain the implementation guidelines. Finally, we will review the benefits of this algorithm.
The work is devoted to the construction and analysis of the gradient method based on a modified explicit second-order Runge--Kutta method, constructed using the Lagrange--Burmann expansion. A two-step method with inertia based on the heavy ball method is proposed. Convergence theorems are proven for strongly convex quadratic and perturbed quadratic functions. Analytical expressions for the optimal parameters of the method are obtained. For the quadratic function it is demonstrated, that the proposed method converges faster, than other well-known accelerated methods. The results of the application of the method to the numerical solution of linear and nonlinear boundary value problems (Dirichlet problem for a 3D Poisson equation, problems from the calculus of variations, problems for integro-differential equations) are presented. It is demonstrated that, in comparison with well-known methods, the proposed method allows one to obtain a numerical solution with the required accuracy for different grid resolutions in a smaller number of iterations and time.
In various fields, artificial intelligence (AI) models are increasingly used for decision-making based on machine learning. In metric machine learning methods, objects are treated as precedents, and only one operation is used: determining the similarity (difference) between these precedents and an unknown object. The main limitation of existing metric methods is related to representing a common feature space for all objects and, consequently, a single measure for measuring distances between objects. This limitation is overcome by constructing a unique local feature space for each object and finding individual measures that determine the hierarchy of its similarity to other objects, relevant to the given context. The article discusses the problem of analyzing sets of objects with local descriptions and proposes a solution using d(S)-metrics, which reflect differences in distance series between the same objects but in different local spaces. Introducing d(S)-metrics allows for further visual analysis of compositions of objects with local descriptions using multidimensional metric scaling. The article provides a practical example of such analysis in the task of recognizing vehicle types based on geometric features of their silhouettes.
The problem of synthesizing an adaptive robust controller using a tensor representation of a quasi-linear parameter-varying model is considered. This type of approximation of nonlinear dynamical systems can be reduced to convex polytopic forms in over a given parameter range. As a result, convex programming methods are applicable to them, including formulation of the problem in terms of linear matrix inequalities. The theory of H-infinity-optimal control is used as a technique for the synthesis of robust controllers. The result of the synthesis is a tensor model that describes a family of robust controllers that ensure stability and controllability of the object for the entire range of varied parameters.
The article reveals the effect of the influence of pairwise non-intersecting and not passing through the origin of coordinates of non-isolated singularities in the lowest coefficient of the generalized Cauchy-Riemann equation on the formulation of boundary value problems. The condition at the boundary of the region turned out to be insufficient to solve such problems. Therefore, we considered a case that combines elements of the Riemann-Hilbert problems on the boundary of a domain and linear conjugation on the surrounding area.
n this paper, we study a modified Hegselmann-Krause model of opinion dynamics based on the bounded confidence principle. This model is formulated as a discontinuous and nonlinear dynamical system. At any time moment of the process of opinion formation, the operator of forming the next opinion of an agent is two-step; first, one takes the average of opinions of agents sharing similar opinions plus his/her own; in the second step, a regularization procedure is performed. A new regularization procedure is applied. We find conditions under which every trajectory tends to a fixed point of the system and study stability properties of fixed points.
The mathematical model of classical enzyme-substrate Michaelis-Menten reaction representing the Cauchy problem for two nonlinear differential equations written in dimensionless form is studied. This system of equations belongs to the class of Tikhonov systems since one of the two equations is singularly perturbed. To obtain an approximate solution we use the holomorphic regularization method. The holomorphic regularization method is a logical continuation of the regularization method of S.A. Lomov and unlike other methods that deduce approximations in the form of series that converge asymptotically allows to obtain solutions to nonlinear singularly perturbed problems in the form of series in powers of small parameter that converge in the usual sense. The exposition of holomorphic regularization method for a system of differential equations of Tikhonov type is given. Deduced approximation to the solution of the enzyme-substrate reaction differential equations system is given by uniform formulas both in the boundary layer and outside it. The advantage of using the holomorphic regularization method is obtaining the formulas for approximate solution, which allow to analyze the approximate solution of an enzymatic reaction over the entire time interval under consideration including the boundary layer. Represented plots of the dependence of substrate concentration and enzyme-substrate complex concentration on the time demonstrate the high accuracy of the obtained approximate solutions even for relatively large values of the small parameter. The obtained approximate solution of the system of enzymatic reaction equations is used for deducing the reaction rate of the substrate and the reaction rate of the enzyme-substrate complex that are valid both in the boundary layer and outside it.
The problem of controlling interorbital flight in the spatial case using the first integrals is considered. Three types of synergistic regulators are proposed. It is shown that the thrust decreases exponentially over time. The executable models of the regulators are designed. The influence of the parameter of the aggregated variable on the geometric shape of the transition orbit is numerically considered. The effect of the choice of an aggregated variable on the amount of work and the amount of thrust. The effect of the angle between the initial and final orbits on the work of transition from orbit to orbit.
This work is an extension of the authors’ earlier investigations into a nonlinear theory of propagation of impurities in solids. Two mathematical models governed by partial differential equations are studied in this paper. Both models describe the dynamics of the impurities’ propagation in a semi-infinite prismatic solid with an impenetrable side surface. The difference between the models is associated with the impurity source that is located at the boundary of the solid. In the first case the source is assumed to be present during a finite time interval, whereas in the second case the impurity source is assumed to be stationary and provide a constant impurity flow. An approach is presented in this paper that allows to transform the nonlinear partial differential equations into nonlinear ordinary differential equations and derive solutions to the latter. A methodology is presented for the formulation of the initial conditions for the nonlinear ordinary differential equations and criteria are formulated for evaluation of the correctness of those conditions. The obtained solutions provide information regarding the time-dependent concentration of the impurities and their finite propagation speed. It is also shown in the paper that the proposed solution approach is also efficient in application to the linear theory of diffusion.
The paper considers a linear discrete-time system operating in a repetitive mode to track a reference trajectory with a given accuracy. The system parameters are incompletely known and are described by the affine uncertainty model. A new iterative learning control design method based on information about the measured output signal is obtained; this method takes into account the saturation-type nonlinearity inherent in the actuators of robots and allows achieving the required accuracy. The problem statement is motivated by the development trends of high-precision smart and additive manufacturing as well as medical rehabilitation robots. An example illustrates the effectiveness of this method.
The paper addresses the filtering a continuous-time Markov chain states that can be observed through linear measurements perturbed by a Wiener process. There is supposed the presence of uncertainty in the intensity of measurements noises. The problem is worked out under the assumption of unknown intensity but subject to its known upper bound. If there is no uncertainty in measurements the optimal solution is provided by the Wonham filter that doesn't ensure stable numerical implementations. The paper exposes that the Wonham filter shows robustness in the presence of uncertainty if model's parameters don't imply its divergent. It is detected that to cope with divergence tracking and handling trajectories aren't sufficient in the case of uncertainty. The more efficient way is to consider discretized approximations of the Wonham filter implemented for a discrete model that approximates the initial continuous-time measurements system. Such an approach perceptibly advantages if numerical implementations contain divergent trajectories. If there are no divergent trajectories, then the discretized filters give a slightly worse result but acceptable.
The paper is addressed to the permanence problem for a generalized Lotka-Volterra system modeling interaction of species in a biological community. The impact of a constant delay and switching of parameters on the dynamics of the system is taken into account. Our analysis is based on the Lyapunov direct method. An original construction of a Lyapunov--Krasovskii functional is proposed. The conditions for the existence of such a functional are formulated in terms of feasibility of special systems of linear algebraic inequalities. It is proved that, under these conditions, the investigated system is permanent for any constant positive delay and any admissible switching signal. In addition, a discrete-time counterpart of the considered model is studied for which the permanence analysis is fulfilled, as well. A comparison of the obtained permanence conditions with known ones is provided. It is shown that the constraints on the system parameters derived in this paper are less conservative.
The article is devoted to the development of a new approach to the series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process. This approach was proposed by the author in 2022 and is based on generalized multiple Fourier series in complete orthonormal systems of functions in Hilbert space. In the previous parts of this work, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 6 were obtained. At that, the expansions were constructed using two specific bases in Hilbert space. More precisely, Legendre polynomials and the trigonometric Fourier basis were used. In this paper, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4 are obtained on the base of arbitrary complete orthonormal systems of functions in Hilbert space. Sufficient conditions for the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity are formulated in terms of trace series. The results of the article will be useful for construction of strong numerical methods with orders 1.0, 1.5 and 2.0 (based on the Taylor-Stratonovich expansion) for Ito stochastic differential equations with non-commutative noise.
The studies carried out in the author's earlier works show the possibility of a stochastic process in the numerical integration of systems of autonomous differential equations (ADE) of the Lorentz type. They also noted that with an increase in the number of equations (degrees of freedom) in the ADE system, the stochastic nature of the process decreases – the "determinization" of the process occurs. In this paper, an attempt is made to characterize the process of transition from one iterative step to another (when integrating the ADE system by numerical methods) by some average value of the time interval for all the equations of the system, and then to determine the relationship between this value and the change in the entropy of the system. Two limiting cases are found when describing such a system: "classical" - when time monotonically increases during the transition from one iterative step to another, and "quantum" - when time is defined ambiguously and a violation of cause-and-effect relationships in the system is possible.
We discuss the recent progress on studying volume contraction for linear cocycles generated by delay equations obtained by the authors. On the basis, we use adapted metrics constructed explicitly or via the Frequency Theorem. In contrast to many existing results, this approach allows to provide effective estimates in terms of the system parameters (including delays). We illustrate the exposed general results by means of the nonautonomous and classical Nicholson blowflies models, where effective dimension estimates for global attractors and robust conditions for the global stability are obtained.
In this paper, a comparative analysis of various approaches to modeling the process of capillary imbibition in oil-saturated rocks is carried out. Today, the permeability of developed oil fields is decreasing, due to which capillary processes begin to make a significant contribution to filtration. Therefore, the issue of reservoir-scale imbibition modeling is becoming increasingly relevant. Capillary imbibition is a process of spontaneous filtration of a liquid into a porous medium under the action of capillary forces. The aim of the study is to analyze approaches to the mathematical description of this process. To do this, one-dimensional models of single-phase imbibition by Handy, Lee and Horn, Benavente and Kai are considered, the Schmid one-dimensional two-phase imbibition equation is solved, and the ability of the filtration model in the tNavigator software package to predict imbibition is checked. For verification, real experiments on core are simulated. Based on the simulation results, conclusions are drawn that single-phase models have an increased error due to the interaction of water with the second phase and non-physical values of free parameters, which is why it is not recommended to scale them to large rock volumes. The two-phase model does not consider gravity, due to which the error increases at the last stages of vertical imbibition, so this approach must be modified to consider gravitational forces in the imbibition equation. A numerical one-dimensional experiment on a hydrodynamic simulator using the proposed core modeling technique showed the best convergence with experimental data with an error of 1 to 4%, which is why it is recommended for modeling capillary processes on the scale of a well.
The dynamics of the chaotic Rossler with external force the phase of which depends on the state of the system is investigated. The study was carried out with the method of charts of Lyapunov exponents that identify areas of different types of dynamics on the plane frequency - amplitude of the external force. Their transformation is discussed with an increase in the parameter responsible for the dependence of the phase on the dynamical variable. The possibility of quasi-periodic dynamics with the set of Arnold tongues was demonstrated. It is shown that with a strong dependence of the phase on the variable of the oscillator, a picture of regular tongues of periodic regimes embedded into the region of chaotic dynamics is formed. Tongues have a threshold in the value of the amplitude of external force. Inside the tongues there are several period-doubling bifurcations of the limit cycles and quasi-periodic dynamics are possible.
The publication presents the problems of identifying data types (typesetting) and semantic description of the attributes when managing structured data and master data (Master Data Management). A formal definition of the generalized attribute typesetting problem is given, which allows generation of the additional data types. This problem allows using the discrete Bellman optimality principle under special criteria of the target function. A unified architecture of the deep generative neural network addressing simultaneously the generalized attribute typesetting and semantic description generation problems is proposed. The architecture is based on the generative adversarial autoencoder architecture (AAE) using the mechanisms of soft-attention, and long-term memory (SCRN). The effectiveness of such implementation, in particular, is achieved through the application of the principles of dynamic programming within each epoch of the network training.
The publication presents an approach to the use of discrete optimization algorithms, in particular, the search for suboptimal solutions. The theory of enumerators, proposed by the famous Leningrad mathematician I.V. Romanovsky, and the operation of their summation, which is proposed to be used to create multi-domain suboptimal algorithms,are considered. The paper presents an efficient algorithm to sum enumerators based on the recalculation of the Pareto boundary. Motivations for using the proposed algorithm within the framework of a well–known task in the field of Master Data Management are given.
he problem of controlling the interorbital maneuvers of a point in a gravitational field is considered. Using the first integrals of the system, a fairly simple synergetic regulator was designed. With its help, the problem of choosing the optimal point of departure from the initial orbit is investigated. At the same time, the maximum value of the thrust generated by the control is selected as a criterion. The problem of transition from the initial to the final orbit in the case of orbits touching is considered. The question of choosing a parameter defining the control vector is investigated.
