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Abstract: A method of asymptotic integration based on application of a generalized Prüfer transformation is elaborated for a class of Emden–Fowler type equations, and a relationship with two-scale expansion method is established. PubDate: 2022-03-21
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Abstract: A comparative analysis of various projection Monte Carlo algorithms as applied to estimating the particle flow through a layer of medium with scattering of the Henyey–Greenstein type is carried out. The possibility of minimizing the mean-square error by equalizing the corresponding stochastic and deterministic terms is investigated. PubDate: 2022-02-01
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Abstract: We consider the Cauchy problem for a one-dimensional hyperbolic equation. The lowest order coefficient and the right-hand side oscillate in time at a high frequency, and the amplitude of the lowest order coefficient is small. The way of reconstructing these oscillating functions from partial solution asymptotics given at a certain point of the domain is studied. PubDate: 2022-02-01
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Abstract: The dynamics of chains of coupled logistic equations with delay are studied using methods of local analysis. It is shown that the critical cases have infinite dimension. As the main results, special nonlinear boundary value problems of the parabolic type describing the evolution of solutions to the initial equation that slowly oscillate at the equilibrium state are constructed. PubDate: 2022-02-01
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Abstract: The dynamics of systems in a potential field is considered in the case where kinetic energy can be represented in conformal form. In the case of two degrees of freedom, conformal coordinates always exist. It is assumed that the system can additionally experience gyroscopic forces. For a fixed value of the total energy, the equations of motion are reduced to a simple “universal” form by changing time. A case of integrability of the equations of motion at fixed total energy is indicated. PubDate: 2022-02-01
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Abstract: An existence and uniqueness theorem for a classical solution to the system of equations describing thermal boundary layers in viscous media with the Ladyzhenskaya rheological law is generalized. PubDate: 2022-02-01
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Abstract: The paper studies the asymptotic behavior of the optimal control for the Poisson type boundary value problem in a domain perforated by holes of an arbitrary shape with Robin-type boundary conditions on the internal boundaries. The cost functional is assumed to be dependent on the gradient of the state and on the usual L2-norm of the control. We consider the so-called “critical” relation between the problem parameters and the period of the structure \(\varepsilon \to 0\) . Two “strange” terms arise in the limit. The paper extends, by first time in the literature, previous papers devoted to the homogenization of the control problem which always assumed the symmetry of the periodic holes. PubDate: 2022-02-01
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Abstract: The problem of controlling oscillations near the equilibrium position of a scleronomic mechanical system with several degrees of freedom is solved. One degree of freedom is not controllable directly, while the others are controlled by servos. An original method for finding an optimal control of the oscillation amplitude for the uncontrolled degree of freedom by choosing a control law for the other degrees of freedom is proposed. The set of controlled coordinates can include both positional and cyclic coordinates. Compared to Pont-ryagin’s maximum principle, the proposed method does not contain adjoint variables and significantly reduces the dimension of the analyzed system of differential equations. The effectiveness of the method is demonstrated as applied to a specific pendulum system. PubDate: 2022-02-01
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Abstract: We study the probability threshold for the property of strong colorability with a given number of colors of a random \(k\) -uniform hypergraph in the binomial model \(H(n,k,p)\) . A vertex coloring of a hypergraph is said to be strong if any edge does not have two vertices of the same color under it. The problem of finding a sharp probability threshold for the existence of a strong coloring with q colors for \(H(n,k,p)\) is studied. By using the second moment method, we obtain fairly tight bounds for this quantity, provided that q is large enough in comparison with k. PubDate: 2022-02-01
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Abstract: In this work, we study nontrivial generalizations of Ramsey numbers to the case of arbitrary sequences of graphs. For many classes of sequences, exact values or asymptotics of Ramsey numbers are found. PubDate: 2022-02-01
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Abstract: We study the stability of equilibrium in the problem known as “a ball on a rotating saddle,” which was first considered by the famous Dutch mathematician Brauer in 1918. He showed that, in the case of a smooth surface, the saddle point, unstable in the absence of rotation, can be stabilized in a certain range of angular velocities. Later, this system was considered by Bottema from a standpoint of bifurcation theory. The physical analogue of this problem is the Nobel Laureate Paul’s ion trap: here, the rotating solid support is replaced by a quadrupole with a periodically changing voltage and gravity is replaced by an electrostatic field. The stability conditions were obtained in a linear approximation, and their sufficiency has not yet been proven. In this paper, such a proof is carried out by methods of Hamiltonian mechanics. PubDate: 2021-11-01
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Abstract: An object t moving in \({{\mathbb{R}}^{3}}\) goes around a solid convex set along the shortest path \(\mathcal{T}\) under observation. The task of an observer f (moving at the same speed as the object) is to find a trajectory closest to \(\mathcal{T}\) that satisfies the condition \(\delta \leqslant {\text{ }}f - t{\text{ }} \leqslant K\delta \) for a given \(\delta > 0\) . This condition makes it possible to track the object along the entire trajectory \(\mathcal{T}\) . A method is proposed for constructing an observer trajectory that ensures that the indicated inequality holds with a constant \(K\) arbitrarily close to unity and the object can be observed on its trajectory \(\mathcal{T}\) , except for an arbitrarily small segment of \(\mathcal{T}\) . PubDate: 2021-11-01
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Abstract: We introduce a family \({{r}_{\lambda }},\lambda \in \mathbb{C}\) of complex-valued stochastic processes making it possible to construct a probabilistic representation for the resolvent of the operator \( - \frac{1}{2}\frac{{{{d}^{2}}}}{{d{{x}^{2}}}}\) . For \(\lambda = 0\) the process \({{r}_{\lambda }}\) is real-valued and coincides with the Brownian local time process. PubDate: 2021-11-01
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Abstract: For an aggregated quasi-gasdynamic system of equations for a homogeneous gas mixture, we give an entropy balance equation with a nonnegative entropy production in the presence of diffusion fluxes. We also derive the existence, uniqueness, and L2-dissipativity of weak solutions to an initial-boundary value problem for the system linearized at a constant solution. Additionally, the Petrovskii parabolicity and local-in-time classical unique solvability of the Cauchy problem for the quasi-gasdynamic system itself are established. PubDate: 2021-11-01
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Abstract: For a system of electrodynamic equations corresponding to time-periodic oscillations, two inverse problems of determining anisotropic conductivity from given phaseless information on solutions of some direct problems are considered. It is supposed that the conductivity is described by a diagonal matrix \(\sigma (x) = {\text{diag}}({{\sigma }_{1}}(x),{{\sigma }_{2}}(x),{{\sigma }_{3}}(x))\) such that \(\sigma (x) = 0\) outside of a compact domain Ω. Plane waves coming from infinity are considered impinging on the inhomogeneity. To determine the unknown functions, the moduli of some components of the electric intensity vector of the total or scattered high-frequency electromagnetic fields are given on the boundary of Ω. It is proved that this information reduces the inverse problems to problems of X-ray tomography. PubDate: 2021-11-01
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Abstract: — Some representations of states of quantum systems are discussed, and their equivalence is proved. In particular, an approach going back to L.D. Landau in which the density operator is constructed using a reduction of a pure state of a quantum system described by the tensor product of suitable Hilbert spaces is presented. Under these assumptions, changes in the states of subsystems of a quantum system caused by experiments are investigated. PubDate: 2021-11-01
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Abstract: Abstract—We show that a quotient of a non-trivial Severi–Brauer surface S over arbitrary field \(\Bbbk \) of characteristic 0 by a finite group \(G \subset {\text{Aut}}(S)\) is \(\Bbbk \) -rational if and only if G is divisible by 3. Otherwise, the quotient is birationally equivalent to S. PubDate: 2021-11-01
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Abstract: Tensor invariants (differential forms) for homogeneous dynamical systems on tangent bundles of smooth two-dimensional manifolds are presented. The connection between the presence of these invariants and the full set of first integrals necessary for the integration of geodesic, potential, and dissipative systems is shown. The force fields introduced into the considered systems make them dissipative with dissipation of different signs and generalize previously considered force fields. PubDate: 2021-11-01
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Abstract: Abstract—In this note we study the stationary Kolmogorov equation and prove that, in the case where the diffusion matrix satisfies Dini’s condition and the drift coefficient is locally integrable to a power greater than the dimension, the ratio of two probability solutions belongs to the Sobolev class, and in the case of existence of a Lyapunov function or the global integrability of the coefficients with respect to the solution a probability solution is unique. PubDate: 2021-11-01
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Abstract: A method is proposed for constructing a CABARET scheme that approximates a hyperbolic system of conservation laws that cannot be written in the form of invariants. This technique is based on the method of quasi-invariants and additional flux correction, which ensures monotonization of the difference solution in calculating discontinuous solutions with shock waves and contact discontinuities. As an example, a system of conservation laws for nonisentropic gas dynamics with a polytropic equation of state is considered. Test calculations of the Blast Wave initial-boundary value problem showed that the proposed scheme suppresses nonphysical oscillations leading to the instability of the difference solution in the case when the CABARET scheme is used without additional flux correction. PubDate: 2021-11-01
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