G-analysis of high-dimensional observations

Consistent and asymptotically normal G-estimators are obtained for generalized variance and normalized spectral function of the covariance matrix when Kolmogorov's condition is satisfied. The proofs are based on the application of limit theorems for random determinants and resolvents of random matrices.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 115–121, 1986.     

Basic boundary-value problems for one equation with fractional derivatives

We prove certain properties of solutions of the equation

Basic asymptotic expansions of solutions to the sixth Painlevé equation

Asymptotic expansions of solutions of three types are found for the sixth Painlevé equation in its three singular points. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

Ill-posedness of Kawahara equation and Kaup-Kupershmidt equation

In this paper we consider the Cauchy problems for the Kawahara equation and the Kaup-Kupershmidt equation. By using the general well-posedness principle introduced by I. Bejenaru and T. Tao (2006) [1], we prove that the Kawahara equation is ill-posed for the initial data in Hs(R) with and the Kaup-Kupershmidt equation is ill-posed for the initial data in Hs(R) with .     

Two-dimensional transport equation as Fredholm integral equation

The transport equation has many applications in various fields of science and engineering. In this paper we shown that we can transform a transport equation in two-dimensional case into a Fredholm integral equation of the second kind with a compact integral operator for the angular flux by using the Sumudu transform.     

Exact solutions to difference equation models of Burgers' equation

This article gives exact solutions to several difference equation models of Burgers' equation. The particular cases considered correspond to the diffusion-free, nonlinear steady-state and linear steady-state situations.     

From the main equation to the Klein-Gordon equation

The purpose of this paper is to bring out some connections between the work of Isaacs and physics. The first five sections are devoted to classical dynamics. The theories of Hamilton and Jacobi are revisited via the geometrical approach to optimal control and de Broglie's ideas. The other sections contain a stochastic-type generalization, leading to the Klein-Gordon equation of relativistic wave mechanics.     

Equivalence of the Navier-Stokes equation and the infinite-dimensional Burgers equation

In this paper, we prove the equivalence of the Navier-Stokes equation and the infinite-dimensional Burgers-type equation with the Laplace-Lévy operator. An explicit formula for the solution of a certain system of linear equations arising in studying the circulation of the solution of the Navier-Stokes equation is presented. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 109–120, 2006.     

Derivation of quantum kinetic equation from the Liouville-von Neumann equation

A quantum Markov kinetic equation is derived from the Liouville—von Neumann equation in the framework of second-order kinetic perturbation theory. The proposed method can be used to obtain an equation that describes the quantum diffusion of a particle in a crystal.Moscow Aviation Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No.1, pp. 33–43, July, 1994.     

The hyperstability of general linear equation via that of Cauchy equation

Aequationes mathematicae - In this paper, we show that the hyperstability of the general linear equation recently proved by Piszczek (Aequationes Math 88:163–168, 2014) is a direct...     

Generalization of the Maxwell equation and relation to the elastic equation

In this paper, we propose a hyperbolic system of first‐order pseudo‐differential equations as generalization of the Maxwell equation. We state basic properties of this system corresponding to the ones of the (usual) Maxwell equation and explain that several known generalized Maxwell equations presented by some researchers can be integrated into the system. Namely, their equations can be regarded as our equation in special cases. Their generalized equations admit not only transversal but also longitudinal waves and are examined from the physical viewpoint. Using the present system, from the mathematical viewpoint, we interpret the meaning for presence of the longitudinal wave (with the transversal one) in their generalized equations. This presence means existence of more than one non‐zero characteristic root for the system (ie, non‐zero eigenvalue of the symbol). We prove also that our system becomes a first‐order expression of (generalized) elastic equations. Furthermore, it is shown that introducing the elastic equations implies expressing the generalized Maxwell equations by the potentials.     

Heat equation and wave equation with general stochastic measures

We consider the heat equation and wave equation with constant coefficients that contain a term given by an integral with respect to a random measure. Only the condition of sigma-additivity in probability is imposed on the random measure. Solutions of these equations are presented. For each equation, we prove that its solutions coincide under certain additional conditions. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1675 – 1685, December, 2008.     

On reducing the Heun equation to the hypergeometric equation

The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross-ratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higher-degree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations. (SIAM J. Math. Anal. 10 (3) (1979) 655).     

Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method

The modified simple equation method is employed to find the exact solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov （KPP） equation. When certain parameters of the equations are chosen to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

Boundary conditions of the second-order differential equation and the Riccati equation

The conversion of a second-order linear ordinary differential equation with variable coefficients into a Riccati equation depends on whether the second-order problem is an initial-value or two-point boundary-value problem. The distinction is critical in determining the initial condition for the Riccati equation. If the second-order problem is an initial-value problem, the choice of the Riccati transformation depends on whether a zero initial condition for the function or its derivative is specified. If the problem is a two-point boundary-value problem, special methods must be introduced as described in the paper.