The first initial boundary value problem for a compound type equation in a three-dimensional multiply connected region

The method of boundary integral equations is used for solving the first initial boundary value problem for a compound type equation in a three-dimensional multiply connected region. The problem is reduced to a uniquely solvable integral equation. The solution of the problem is obtained in the form of dynamic potentials whose density satisfies this integral equation. Thus the existence theorem is proved. Moreover, the uniqueness of the solution is also studied. All the results are valid for both interior and exterior regions provided that the corresponding conditions at infinity are taken into account. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 249–265, August, 2000.     

Exact elliptic compactons in generalized Korteweg–De Vries equations

Using the action principle, and assuming a solitary wave of the generic form u(x,t) = AZ(β(x + q(t)), we derive a general theorem relating the energy, momentum, and velocity of any solitary wave solution of the generalized Korteweg‐De Vries equation K*(l,p). Specifically we find that , where l,p are nonlinearity parameters. We also relate the amplitude, width, and momentum to the velocity of these solutions. We obtain the general condition for linear and Lyapunov stability. We then obtain a two‐parameter family of exact solutions to these equations, which include elliptic and hyper‐elliptic compacton solutions. For this general family we explicitly verify both the theorem and the stability criteria. © 2006 Wiley Periodicals, Inc. Complexity 11: 30–34, 2006     

The viability property of controlled jump diffusion processes

In this paper, we first give a comparison theorem of viscosity solution to some nonlinear second order integrodifferential equation. And then using the comparison theorem, we obtain a necessary and sufficient condition for the viability property of some controlled jump diffusion processes which can keep the solution within a constraint K.     

An infinite-dimensional version of the Borsuk-Ulam theorem

We study the solvability of the equation a(x) = f(x) on a sphere in a Banach space, where a is a closed surjective linear operator and f is an odd a-compact map. We also estimate the topological dimension of the solution set and give applications of the corresponding theorem to some problems in differential equations and other fields of mathematics.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 1–5, 2004Original Russian Text Copyright © by B. D. GelmanSupported by the Russian Foundation for Basic Research, Grant No. 02-01-00189.     

Stability of solutions of an equation arising in hydromechanics

We study the strong stability of the equation describing small oscillations of an elastic pipe conveying an ideal fluid. We prove the existence and uniqueness theorem for the generalized solution and find sufficient conditions for strong stability in the case of a pulsing fluid. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 15–24, January, 2000.     

Problem of determining the nonstationary potential in a hyperbolic-type equation

We solve the problem of determining the hyperbolic equation coefficient depending on two variables. Some additional information is given by the trace of the direct problem solution on the hyperplane x = 0. We estimate the stability of the solution of the inverse problem under study and prove the uniqueness theorem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 220–225, August, 2008.     

On the number of summands in the asymptotic formula for the number of solutions to Waring's equation

In the paper, an estimate of the number of summands in the asymptotic formula for the number of solutions to Waring's equation is obtained. This is achieved by means of a recurrent process leading to a greater reduction than that in Vinogradov's mean value theorem.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 285–296, August, 1998.The author wishes to thank N. M. Korobov for setting the problem and supervising his work.     

A main theorem of spectral relationships for Volterra–Fredholm integral equation of the first kind and its applications

This paper is concerned to derive the main theorem of spectral relationships of Volterra–Fredholm integral equation (V‐FIE) of the first kind in the space L₂[?1,1]×C[0,T], ?1?x?1, 0?t?T<1. The Fredholm integral (FI) term is considered in position and its kernel takes a logarithmic form multiplying by a continuous function. While Volterra integral (VI) term in time with a positive continuous kernel. Many important special and new cases can be established from the main theorem. Moreover, we use it to solve V‐FIE of the second kind in the same space. The numerical results are computed and the error is calculated using Maple 12. Copyright © 2010 John Wiley & Sons, Ltd.     

Differential equations with almost periodic or conditionally periodic coefficients: Recurrence and reducibility

Recurrence of an individual trajectory of a linear nonautonomous differential equation on a compact Lie group for the case of an almost periodic or conditionally periodic dependence of the right-hand side on time is proved. The relation between recurrence and reducibility is examined.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 229–237, August, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00378 and No. 96-15-96072.     

On a boundary value problem for a p‐Dirac equation

The p‐Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p‐Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p‐Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p‐Laplace equation into the p‐Dirac equation. This equation will be solved iteratively by using a fixed‐point theorem. Applying operator‐theoretical methods for the p‐Dirac equation and p‐Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.     

Some conic bundless

A representation of the anticanonical K3 surface of a singular pencil of conics is described. This generalizes the well-known Shokurov theorem.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 903–910, June, 1998.The author is greatly indebted to V. A. Iskovskikh, Yu. G. Prokhorov, and I. A. Chel'tsov for fruitful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00820 and by INTAS under grant No. 93-2805.00-00-00.     

Asymptotic behavior of large solution to elliptic equation of Bieberbach–Rademacher type with convection terms

We analyze the asymptotic behavior of solutions to nonlinear elliptic equation Δu±|u|^q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)=∞ as , where Ω is a smooth bounded domain in R^N, for some , and . Our approach employs Karamata regular variation theory combined with the method of lower and supper solution.     

New upper bounds on the decomposability of planar graphs

It is known that a planar graph on n vertices has branch‐width/tree‐width bounded by . In many algorithmic applications, it is useful to have a small bound on the constant α. We give a proof of the best, so far, upper bound for the constant α. In particular, for the case of tree‐width, α < 3.182 and for the case of branch‐width, α < 2.122. Our proof is based on the planar separation theorem of Alon, Seymour, and Thomas and some min–max theorems of Robertson and Seymour from the graph minors series. We also discuss some algorithmic consequences of this result. © 2005 Wiley Periodicals, Inc. J Graph Theory     

The law of the iterated logarithm for the solution of the Burgers equation with random initial data

The law of the iterated logarithm is established for the solution of the one-dimensional Burgers equation in the case where the initial potential is described by a zero-range shot noise.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 812–823, December, 1998.The author wishes to thank Professor A. V. Bulinskii for setting the problem and for his attention to the work on the paper.     

A convergence theorem for the fast multipole method for 2 dimensional scattering problems

The Fast Multipole Method (FMM) designed by V. Rokhlin rapidly computes the field scattered from an obstacle. This computation consists of solving an integral equation on the boundary of the obstacle. The main result of this paper shows the convergence of the FMM for the two dimensional Helmholtz equation. Before giving the theorem, we give an overview of the main ideas of the FMM. This is done following the papers of V. Rokhlin. Nevertheless, the way we present the FMM is slightly different. The FMM is finally applied to an acoustic problem with an impedance boundary condition. The moment method is used to discretize this continuous problem.

     

Local analytic solutions of a functional differential equation

In this paper, we are concerned with the existence of analytic solution of a functional differential equation αz+βx^′(z)=x(az+bx^″(z)), where are four complex numbers. We first discuss the existence of analytic solutions for some special cases of the above equation. Then, by reducing the equation with the Schröder transformation to the another functional equation with proportional delay, an existence theorem is established for analytic solutions of the original equation. For the constant λ given in the Schröder transformation, we discuss the case 0<|λ|<1 and λ on the unit circle S¹, i.e., |λ|=1. We study λ is at resonance, i.e., at a root of the unity and λ is near resonance under the Brjuno condition.     

Singular Perturbation Elliptic Boundary-Value Problems in the Case of a Nonisolated Root of the Degenerate Equation

For a singularly perturbed elliptic equation (the Neumann boundary-value problem), we prove a theorem on the passage to the limit for the case in which the degenerate equation has a nonisolated root.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 26–36.Original Russian Text Copyright © 2005 by V. F. Butuzov, M. A. Terent’ev.     

On Jordan ideals and matrix rings

For A, a commutative ring, and results by Costa and Keller characterize certain -normalized subgroups of the symplectic group, via structures utilizing Jordan ideals and the notion of radices. The following work creates a Jordan ideal structure theorem for -graded rings, A₀A₁, and a -graded matrix algebra. The major theorem is a generalization of Costa and Keller’s previous work on matrix algebras over commutative rings.     

New results on the existence of periodic solutions for a higher‐order Li énard type p‐Laplacian differential equation

In this paper, by using the continuation theorem of coincidence degree theory, we consider the higher‐order Li énard type p‐Laplacian differential equation as follows Some new results on the existence of periodic solutions for the previous equation are obtained, which generalize and enrich some known results to some extent from the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

The cauchy problem for a nonlinear integro-differential transport equation

In the present paper, we study the Cauchy problem for a nonlinear time-dependent kinetic neutrino transport equation. We prove the existence and uniqueness theorem for the solution of the Cauchy problem, establish uniform bounds int for the solution of this problem, and prove the existence and uniqueness of a stationary trajectory and the stabilization ast→∞ of the solution of the time-dependent problem for arbitrary initial data. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 677–686, May, 1997. Translated by A. M. Chebotarev