We consider the problem of realization of hypergraphs on a graph provided each hyperedge is realized by a subgraph in which exactly two vertices have odd degree. This problem is related to Cycle Double Cover conjecture. We prove that checking the existence of realization is computationally hard. The hardness is proved in various settings: for realizations on all graphs, on simple graphs, and on graphs from several restricted classes.
For the first time, the problem of optimal tracking with a given reference trajectory and an integral quadratic performance criterion in the presence of singular perturbations is considered. The decomposition method is used to analyze singularly perturbed differential systems that arise in solving this problem. The method is based on the technique of integral manifolds of fast and slow motions. A suboptimal control is constructed the use of which leads to a difference in the values of the minimized functional for the optimal and suboptimal controls by an amount of the order of the second power of a small parameter characterizing singular perturbations.
We use the OpnFoam package to numerically study the turbulent flow over a wavy surface for various values of the amplitude and wavelength of the perturbation of the channel wall. The RANS and LES models are used to describe turbulent characteristics. The Reynolds number in the flow is 20 000. The average profiles of velocities and shear stresses on the channel wall are obtained. The values of the amplitude and phase shift for perturbations of the shear stress are calculated for various geometrical parameters of the channel. Comparison with the theoretical model and experimental results is carried out.
A continuous–discrete stochastic model is presented that describes the dynamics of the number of susceptible and infectious individuals visiting a certain facility. The individuals enter the facility both separately and as part of groups of individuals arranged according to some characteristics. The duration of stay of individuals on the territory of the facility is specified using distributions other than exponential. Individuals who entered the facility as part of a certain group leave the facility as part of the same group. Infectious individuals spread viral particles contained in the airborne mixture they secrete. A certain amount of the airborne mixture containing viral particles settles on the surfaces of various objects in places of the facility that are generally accessible to individuals. The area of the infected surface (the surface containing the settled airborne mixture with viral particles) is described using a linear differential equation with a jumping right-side and discontinuous initial data. Susceptible individuals contacting infectious individuals and contaminated surfaces may be infected. A probabilistic formalization of the model is presented, and an algorithm for numerical simulation of the dynamics of the components of the constructed stochastic process using the Monte Carlo method is described. The results of a numerical study of the expectations of stochastic variables describing the number of contacts of susceptible individuals with infectious ones and with infected surfaces per one susceptible individual for a fixed period of time are presented.
To predict the behavior of reservoir fluids in porous media and their investigation at the macroscopic level, it is necessary to study in detail the hydrodynamic flows in porous media at the microscale level from the point of view of the individual pore spaces taking into account their structural features. This paper deals with the analysis of a periodic flow of a viscous incompressible fluid and dispersed systems in a flat channel of rectangular cross section with irregular side walls under a constant pressure drop. Using an efficient numerical approach based on the 3D Boundary Element Method accelerated by the Fast Multipole Method on heterogeneous computing architectures, the influence of irregularities of various size and shape present on microchannel walls on the hydrodynamic flows of the viscous fluid flow and the emulsion droplet dynamics in a capillary micromodel of the porous medium is studied. The results of the present paper can also be useful in the design of microfluidic devices.
The article proposes a mathematical model of wastewater treatment in a filter based on the use of biofilm. In this model, microorganisms destroy harmful impurities contained in water. The impurities are “food” for the microorganisms. The filter contains a large number of loading elements. A system of partial differential equations with boundary conditions is given for one loading element, which is a cylindrical rod whose surface is covered with a biologically active film. This system includes a parabolic equation in a three-dimensional domain and a hyperbolic equation on part of the surface of this domain, the equations being related via the boundary condition and the potential in the hyperbolic equation. Further, an asymptotic analysis of this system is carried out, which permits one to reduce the model of an individual element to solving a simple ordinary differential equation; a rigorous mathematical justification of the proposed method is given. Here a mathematical method for constructing asymptotics in so-called “thin domains” is used. The method is a simplification of a complex combined model based on the laws of hydrodynamics and diffusion. We use this as a basis to propose a model of the operation of the entire wastewater treatment device containing a large number (millions) of such elements.
When constructing block ciphers, it is necessary to use vector Boolean functions with special cryptographic properties as S-blocks for the cipher’s resistance to various types of cryptanalysis. In this paper, we investigate the following S-block construction: let ( pi ) be a permutation on ( n ) elements, let ( pi ^i ) be the ( i )-fold application of the permutation ( pi ), and let ( f ) be a Boolean function of ( n ) variables. Define a vector Boolean function ( F_{pi }colon mathbb {Z}_2^n to mathbb {Z}_2^n ) as ( F_{pi }(x) = (f(x), f(pi (x)), ldots , f(pi _{n-1}(x))) ). We study the cryptographic properties of ( F_{pi } ) such as high nonlinearity, balancedness, and low differential ( delta )-uniformity in the dependence on the properties of ( f ) and ( pi ) for small ( n ). Complete sets of Boolean functions ( f ) and vector Boolean functions ( F_{pi } ) of few variables with maximum algebraic immunity are also obtained.
The paper is devoted to solving the problem of scheduling the work of the call center staff. A model of integer linear programming is formulated, the problem is shown to be NP-hard, and a genetic algorithm is proposed that takes into account the specifics of the problem. An experimental comparison of optimal solutions obtained using the CPLEX package with solutions found by the genetic algorithm is carried out. The computational experiment has shown the practically acceptable accuracy of the solutions obtained by the genetic algorithm and its applicability to large-dimensional problems.
We consider a one-dimensional inverse problem of determining the coefficient multiplying a power-law gradient nonlinearity in a semilinear wave equation. Theorems on the existence and uniqueness of the solution of the direct problem and local existence and stability of the solution of the inverse problem are proved.
We study a collision of high-amplitude resonant solitons in a liquid with gas bubbles moving towards each other. The interaction is of an inelastic nature. The amplitude of the state of full merging of two resonant solitons is three times their initial amplitudes. Two intense sonic precursors are emitted during the interaction. After the solitons leave the interaction region, a space-limited zone of coupled bubble pulsations arises in it. These processes reduce the energy of solitons, which never restore to their original state.
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