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The Lipschitz continuous property has been used for a long time to solve the global optimization problem and continues to be used. Here we can mention the work of Piyavskii, Yevtushenko, Strongin, Shubert, Sergeyev, Kvasov and others. Most papers assume a priori knowledge of the Lipschitz constant, but the derivation of this constant is a separate problem. Further still, we must prove that an objective function is really Lipschitz, and it is a complicated problem too. In the case where the Lipschitz continuity is established, Strongin proposed an algorithm for global optimization of a satisfying Lipschitz condition on a compact interval function without any a priori knowledge of the Lipschitz estimate. The algorithm not only finds a global extremum, but it determines the Lipschitz estimate too. It is known that every function that satisfies the Lipchitz condition on a compact convex set is uniformly continuous, but the reverse is not always true. However, there exist models (Arutyunova, Dulliev, Zabotin) whose study requires a minimization of the continuous but definitely not Lipschitz function. One of the algorithms for solving such a problem was proposed by R. J. Vanderbei. In his work he introduced some generalization of the Lipchitz property named $\varepsilon$-Lipchitz and proved that a function defined on a compact convex set is uniformly continuous if and only if it satisfies the $\varepsilon$-Lipchitz condition. The above-mentioned property allowed him to extend Piyavskii’s method. However, Vanderbei assumed that for a given value of $\varepsilon$ it is possible to obtain an associate Lipschitz $\varepsilon$-constant, which is a very difficult problem. Thus, there is a need to construct, for a function continuous on a compact convex domain, a global optimization algorithm which works in some way like Strongin’s algorithm, i.e., without any a priori knowledge of the Lipschitz $\varepsilon$-constant. In this paper we propose an extension of Strongin’s global optimization algorithm to a function continuous on a compact interval using the $\varepsilon$-Lipchitz conception, prove its convergence and solve some numerical examples using the software that implements the developed method.

In this article we consider min-min type of problems or minimization by two groups of variables. In some way it is similar to classic min-max saddle point problem. Although, saddle point problems are usually more difficult in some way. Min-min problems may occur in case if some groups of variables in convex optimization have different dimensions or if these groups have different domains. Such problem structure gives us an ability to split the main task to subproblems, and allows to tackle it with mixed oracles. However existing articles on this topic cover only zeroth and first order oracles, in our work we consider high-order tensor methods to solve inner problem and fast gradient method to solve outer problem.

We assume, that outer problem is constrained to some convex compact set, and for the inner problem we consider both unconstrained case and being constrained to some convex compact set. By definition, tensor methods use high-order derivatives, so the time per single iteration of the method depends a lot on the dimensionality of the problem it solves. Therefore, we suggest, that the dimension of the inner problem variable is not greater than 1000. Additionally, we need some specific assumptions to be able to use mixed oracles. Firstly, we assume, that the objective is convex in both groups of variables and its gradient by both variables is Lipschitz continuous. Secondly, we assume the inner problem is strongly convex and its gradient is Lipschitz continuous. Also, since we are going to use tensor methods for inner problem, we need it to be p-th order Lipschitz continuous ($p > 1$). Finally, we assume strong convexity of the outer problem to be able to use fast gradient method for strongly convex functions.

We need to emphasize, that we use superfast tensor method to tackle inner subproblem in unconstrained case. And when we solve inner problem on compact set, we use accelerated high-order composite proximal method.

Additionally, in the end of the article we compare the theoretical complexity of obtained methods with regular gradient method, which solves the mentioned problem as regular convex optimization problem and doesn’t take into account its structure (Remarks 1 and 2).

In this article a method of legacy information systems handling is offered. During professional activities of specialists of various domains of industry they face with the problem that computer software that was involved in product development stage becomes obsolete much quickly than the product itself. At the same time switch to any modern software might be not possible due to various reasons. This problem is known as "legacy system" problem. It appears when product lifecycle is sufficiently longer than that of software systems that were used for product creation. In this article author offers an approach for solving this problem along with computer application based on this approach.

The paper studies the influence of selective harvest on dynamic modes of the «predator–prey» community with age structure for prey. We use a slight modification of the Nicholson-Bailey model to describe the interaction between predator and prey. We assume the prey population size is regulated by a decrease in survival rate of juvenile with an increase in the size of age class. The aim is to study the mechanisms of formation and evolution of dynamic modes for the structured «predator–prey» community model due to selective harvesting. We considered the cases when a harvest of some part of predator or prey population or one of the prey’s age classes is realized. The conditions of stable coexistence of interacting species and scenarios of the occurrence of oscillatory modes of abundance are studied. It is shown the harvesting of only young individuals of prey or simultaneous removal of young and adult individuals leads to expansion of parameter space domain with stable dynamics of prey population both with and without a predator. At the same time, the bistability domain narrows, in which changing initial conditions leads to the predator either remains in the community or dies from lack of food. In the case of the harvest for prey adult individuals or predator, the predator preservation in the community is ensured by high values of the prey birth rate, moreover bistability domain expands. With the removal of both juvenile preys and predators, an increase in the survival rates of adult prey leads to stabilization of the community dynamics. The juveniles’ harvest can lead to damping of oscillations and stabilize the prey dynamics in the predator absence. Moreover, it can change the scenario of the coexistence of species — from habitation of preys without predators to a sustainable coexistence of both species. The harvest of some part of predator or prey or the prey’s older age class can lead to both oscillations damping and stable dynamics of the interacting species, and to the destruction of the community, that is, to the death of predator.

We found a predominance of α-rhythm patterns in the left hemisphere in healthy people compared to people with COVID-19 history. Moreover, we observe a significant decrease in the left hemisphere contribution to the speech center area in people who have undergone COVID-19 when performing speech tasks.

Our findings show that the signal in healthy subjects is more spatially localized and synchronized between hemispheres when performing tasks compared to people who recovered from COVID-19. We also observed a decrease in low frequencies in both hemispheres after COVID-19.

EEG-patterns of COVID-19 are detectable in an unusual frequency domain. What is usually considered noise in electroencephalographic (EEG) data carries information that can be used to determine whether or not a person has had COVID-19. These patterns can be interpreted as signs of hemispheric desynchronization, premature brain ageing, and more significant brain strain when performing simple tasks compared to people who did not have COVID-19.

In our work, we have shown the applicability of neural networks in helping to detect the long-term effects of COVID-19 on EEG-data. Furthermore, our data following other studies supported the hypothesis of the severity of the long-term effects of COVID-19 detected on the EEG-data of EEG-based BCI. The presented findings of functional activity of the brain– computer interface make it possible to use machine learning methods on simple, non-invasive brain–computer interfaces to detect post-COVID syndrome and develop progress in neurorehabilitation.

Numerical simulation of moving sportsman external flow is presented. The unique method is developed for obtaining integral aerodynamic characteristics, which were the function of the flow regime (i.e. angle of attack, flow speed) and body position. Individual anthropometric characteristics and moving boundaries of sportsman (or sports equipment) during the race are taken into consideration.

Numerical simulation is realized using FlowVision CFD. The software is based on the finite volume method, high-performance numerical methods and reliable mathematical models of physical processes. A Cartesian computational grid is used by FlowVision, the grid generation is a completely automated process. Local grid adaptation is used for solving high-pressure gradient and object complex shape. Flow simulation process performed by solutions systems of equations describing movement of fluid and/or gas in the computational domain, including: mass, moment and energy conservation equations; state equations; turbulence model equations. FlowVision permits flow simulation near moving bodies by means of computational domain transformation according to the athlete shape changes in the motion. Ski jumper aerodynamic characteristics are studied during all phases: take-off performance in motion, in-run and flight. Projected investigation defined simulation method, which includes: inverted statement of sportsman external flow development (velocity of the motion is equal to air flow velocity, object is immobile); changes boundary of the body technology defining; multiple calculations with the national team member data projecting. The research results are identification of the main factors affected to jumping performance: aerodynamic forces, rotating moments etc. Developed method was tested with active sportsmen. Ski jumpers used this method during preparations for Sochi Olympic Games 2014. A comparison of the predicted characteristics and experimental data shows a good agreement. Method versatility is underlined by performing swimmer and skater flow simulation. Designed technology is applicable for sorts of natural and technical objects.

Accurate segmentation of liver plays important in contouring during diagnosis and the planning of treatment. Imaging technology analysis and processing are wide usage in medical diagnostics, and therapeutic applications. Liver segmentation referring to the process of automatic or semi-automatic detection of liver image boundaries. A major difficulty in segmentation of liver image is the high variability as; the human anatomy itself shows major variation modes. In this paper, a proposed approach for computed tomography (CT) liver segmentation is presented by combining exponential entropy and fuzzy c-partition. Entropy concept has been utilized in various applications in imaging computing domain. Threshold techniques based on entropy have attracted a considerable attention over the last years in image analysis and processing literatures and it is among the most powerful techniques in image segmentation. In the proposed approach, the computed tomography (CT) of liver is transformed into fuzzy domain and fuzzy entropies are defined for liver image object and background. In threshold selection procedure, the proposed approach considers not only the information of liver image background and object, but also interactions between them as the selection of threshold is done by find a proper parameter combination of membership function such that the total fuzzy exponential entropy is maximized. Differential Evolution (DE) algorithm is utilizing to optimize the exponential entropy measure to obtain image thresholds. Experimental results in different CT livers scan are done and the results demonstrate the efficient of the proposed approach. Based on the visual clarity of segmented images with varied threshold values using the proposed approach, it was observed that liver segmented image visual quality is better with the results higher level of threshold.

A large number of methods have been developed to solve the Navier – Stokes equations in the case of incompressible flows, the most popular of which are methods with velocity correction by the SIMPLE algorithm and its analogue — the method of splitting by physical variables. These methods, developed more than 40 years ago, were used to solve rather simple problems — simulating both stationary flows and non-stationary flows, in which the boundaries of the calculation domain were stationary. At present, the problems of computational fluid dynamics have become significantly more complicated. CFD problems are involving the motion of bodies in the computational domain, the motion of contact boundaries, cavitation and tasks with dynamic local adaptation of the computational mesh. In this case the computational mesh changes resulting in violation of the velocity divergence condition on it. Since divergent velocities are used not only for Navier – Stokes equations, but also for all other equations of the mathematical model of fluid motion — turbulence, mass transfer and energy conservation models, violation of this condition leads to numerical errors and, often, to undivergence of the computational algorithm.

This article presents an implicit method of splitting by physical variables that uses divergent velocities from a given time step to solve the incompressible Navier – Stokes equations. The method is developed to simulate flows in the case of movable and contact boundaries treated in the Euler paradigm. The method allows to perform computations with the integration step exceeding the explicit time step by orders of magnitude (Courant – Friedrichs – Levy number $CFL\gg1$). This article presents a variant of the method for incompressible flows. A variant of the method that allows to calculate the motion of liquid and gas at any Mach numbers will be published shortly. The method for fully compressible flows is implemented in the software package FlowVision.

Numerical simulating classical fluid flow around circular cylinder at low Reynolds numbers ($50 < Re < 140$), when laminar flow is unsteady and the Karman vortex street is formed, are presented in the article. Good agreement of calculations with the experimental data published in the classical works of Van Dyke and Taneda is demonstrated.

The spatiotemporal dynamics of a three-component model for food web is considered. The model describes the interactions among resource, prey and predator that consumes both species. In a previous work, the author analyzed the model without taking into account spatial heterogeneity. This study continues the model study of the community considering the diffusion of individuals, as well as directed movements of the predator. It is assumed that the predator responds to the spatial change in the resource and prey density by occupying areas where species density is higher or avoiding them. Directed predator movement is described by the advection term, where velocity is proportional to the gradient of resource and prey density. The system is considered on a one-dimensional domain with zero-flux conditions as boundary ones. The spatiotemporal dynamics produced by model is determined by the system stability in the vicinity of stationary homogeneous state with respect to small inhomogeneous perturbations. The paper analyzes the possibility of wave instability leading to the emergence of autowaves and Turing instability, as a result of which stationary patterns are formed. Sufficient conditions for the existence of both types of instability are obtained. The influence of local kinetic parameters on the spatial structure formation was analyzed. It was shown that only Turing instability is possible when taxis on the resource is positive, but with a negative taxis, both types of instability are possible. The numerical solution of the system was found by using method of lines (MOL) with the numerical integration of ODE system by means of splitting techniques. The spatiotemporal dynamics of the system is presented in several variants, realizing one of the instability types. In the case of a positive taxis on the prey, both autowave and stationary structures are formed in smaller regions, with an increase in the region size, Turing structures are not formed. For negative taxis on the prey, stationary patterns is observed in both regions, while periodic structures appear only in larger areas.

The purpose of the study is to simulate the dynamics of traffic flows on city road networks as well as to systematize the current state of affairs in this area. The introduction states that the development of intelligent transportation systems as an integral part of modern transportation technologies is coming to the fore. The core of these systems contain adequate mathematical models that allow to simulate traffic as close to reality as possible. The necessity of using supercomputers due to the large amount of calculations is also noted, therefore, the creation of special parallel algorithms is needed. The beginning of the article is devoted to the up-to-date classification of traffic flow models and characterization of each class, including their distinctive features and relevant examples with links. Further, the main focus of the article is shifted towards the development of macroscopic and microscopic models, created by the authors, and determination of the place of these models in the aforementioned classification. The macroscopic model is based on the continuum approach and uses the ideology of quasi-gasdynamic systems of equations. Its advantages are indicated in comparison with existing models of this class. The model is presented both in one-dimensional and two-dimensional versions. The both versions feature the ability to study multi-lane traffic. In the two-dimensional version it is made possible by introduction of the concept of “lateral” velocity, i. e., the speed of changing lanes. The latter version allows for carrying out calculations in the computational domain which corresponds to the actual geometry of the road. The section also presents the test results of modeling vehicle dynamics on a road fragment with the local widening and on a road fragment with traffic lights, including several variants of traffic light regimes. In the first case, the calculations allow to draw interesting conclusions about the impact of a road widening on a road capacity as a whole, and in the second case — to select the optimal regime configuration to obtain the “green wave” effect. The microscopic model is based on the cellular automata theory and the single-lane Nagel – Schreckenberg model and is generalized for the multi-lane case by the authors of the article. The model implements various behavioral strategies of drivers. Test computations for the real transport network section in Moscow city center are presented. To achieve an adequate representation of vehicles moving through the network according to road traffic regulations the authors implemented special algorithms adapted for parallel computing. Test calculations were performed on the K-100 supercomputer installed in the Centre of Collective Usage of KIAM RAS.