Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.     

Formal asymptotic solution of a singularly perturbed nonlinear optimal control problem

A system of equations that arises in a singularly perturbed optimal control problem is studied. We give conditions under which a formal asymptotic solution exists. This formal asymptotic solution consists of an outer expansion and left and right boundary-layer expansions. A feature of our procedure is that we do nota priori eliminate the control function from the problem. In particular, we construct a formal asymptotic expansion for the control directly. We apply our procedure to a Mayer-type problem. The paper concludes with a worked example.     

Semiclassical asymptotic approximation of the two-dimensional Hartree operator spectrum near the upper boundaries of spectral clusters

We consider an eigenvalue problem for the two-dimensional Hartree operator with a small parameter at the nonlinearity. We obtain the asymptotic eigenvalues and the asymptotic eigenfunctions near the upper boundaries of the spectral clusters formed near the energy levels of the unperturbed operator and construct an asymptotic expansion around the circle where the solution is localized.     

Representation of attraction elements in abstract attainability problems with asymptotic constraints

We consider an abstract attainability problem with asymptotic constraints in a topological space. We construct an extension in the class of ultrafilters of widely interpreted measurable spaces. We study an example of a static problem on the asymptotic attainability in the class of layer functions.     

Asymptotic evaluation of effective complex moduli of fibre‐reinforced viscoelastic composite materials

We propose an asymptotic approach for the evaluation of effective complex moduli of viscoelastic fibre‐reinforced composite materials. Our method is based on the homogenization technique. We start with a non‐trivial expansion of the input plane‐strain boundary value problem by ratios of visco‐elastic constants. This allows to simplify the governing equations to forms analogous to the complex transport problem. Then we apply the asymptotic homogenization method, coming from the original problem on multi‐connected domain to the cell problem, defined on a unit cell of the periodic structure. For the analytical solution of the cell problem we apply the boundary perturbation technique, the asymptotic expansion by a distance between two neighbouring fibres and the method of two‐point Padé approximants. As results we derive uniform analytical representations for effective complex moduli, valid for all values of the components volume fractions and properties.     

Semiclassical asymptotic spectrum of a Hartree-type operator near the upper boundary of spectral clusters

We consider the problem for eigenvalues of a perturbed two-dimensional oscillator in the case of a resonance frequency. The exciting potential is given by a Hartree-type integral operator with a smooth self-action potential. We find asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundary of spectral clusters, which form around energy levels of the nonperturbed operator. To calculate them, we use asymptotic formulas for quantum means.     

Supersymmetry, Witten Complex and Asymptotics for Directional Lyapunov Exponents in Zd

. We study the long distance asymptotic for random walks in random potentials. We use the analytical formulation. We relate the problem to Witten Laplacians via using supersymmetry. We obtain the asymptotic for the directional Lyapunov exponents.     

Determining nodes for semilinear parabolic equations

We are concerned with the determination of the asymptotic behavior of strong solutions to the initial-boundary value problem for general semilinear parabolic equations by the asymptotic behavior of these strong solutions on a finite set. More precisely, if the asymptotic behavior of the strong solution is known on a suitable finite set which is called determining nodes, then the asymptotic behavior of the strong solution itself is entirely determined. We prove the above property by the energy method.     

Generalized elements in problem on asymptotic attainability

We consider a problem which implies the choice of a solution subject to asymptotic constraints. We represent the results as ultrafilters of the space of ordinary estimates (the space is not necessarily endowed with a topology). This representation corresponds to an abstract attainability problem in its nonsequential asymptotic version.     

Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.     

Asymptotic Solution of A Multichannel Scattering Problem with A Nonadiabatic Coupling

We consider a multichannel scattering problem in an adiabatic representation. We assume that the nonadiabatic coupling matrix has a nontrivial value at large internuclear separations, and we construct asymptotic solutions at large internuclear distances. We show that these solutions up to the first order of the perturbation theory are identical to the asymptotic solutions of the reprojection approach, which was previously proposed as a means for solving the electron translation problem in the context of the Born–Oppenheimer method.     

Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.     

On the asymptotics of a solution to an equation with a small parameter at some of the highest derivatives

We study the asymptotic behavior of a solution of the first boundary value problem for a second-order elliptic equation in a nonconvex domain with smooth boundary in the case where a small parameter is a factor at only some of the highest derivatives and the limit equation is an ordinary differential equation. Although the limit equation has the same order as the initial equation, the problem is singulary perturbed. The asymptotic behavior of its solution is studied by the method of matched asymptotic expansions.     

A perturbation method applied to the buckling of a compressed elastica

We consider the problem of carrying out an asymptotic analysis for the phenomenon of bifurcation which occurs at critical values of an axial force applied to an elastic column. In the present setting a discontinuous coefficient precludes the possibility of carrying out the usual asymptotic analysis. The problem is overcome via a nonlinear change of independent variables.     

Computing the asymptotic worst-case of bin packing lower bounds

This paper addresses the issue of computing the asymptotic worst-case of lower bounds for the Bin Packing Problem. We introduce a general result that allows to bound the asymptotic worst-case performance of any lower bound for the problem and to derive for the first time the asymptotic worst-case of the well-known bound L₃ by Martello and Toth. We also show that the general result allows to easily derive the asymptotic worst-case of several lower bounds proposed in the literature.     

Multiplicative asymptotics of solutions of the first boundary value problem on a half-axis for a parabolic equation with a small parameter

In this work we consider the first boundary value problem for a parabolic equation of second order with a small parameter on a half-axis (i.e., we consider the one-dimensional case). We take the zero initial condition. We construct the global (that is, the caustic points are taken into account) asymptotics of a solution for the boundary value problem. The asymptotic solution of this problem has a different structure depending on the sign of the coefficient (the drift coefficient) at the derivative of first order at a boundary point. The constructed asymptotic solutions are justified.     

Existence and stability of periodic contrast structures in the reaction–advection–diffusion problem in the case of a balanced nonlinearity

We study a singularly perturbed periodic problem for the parabolic reaction–advection–diffusion equation with small advection. We consider the case in which there exists an internal transition layer under the conditions of balanced nonlinearity. An asymptotic expansion of the solution is constructed. To substantiate this asymptotics, we use the asymptotic method of differential inequalities. The Lyapunov asymptotic stability of the periodic solution is analyzed.     

Inner and outer estimates for the solution sets and their asymptotic cones in vector optimization

We use asymptotic analysis to develop finer estimates for the efficient, weak efficient and proper efficient solution sets (and for their asymptotic cones) to convex/quasiconvex vector optimization problems. We also provide a new representation for the efficient solution set without any convexity assumption, and the estimates involve the minima of the linear scalarization of the original vector problem. Some new necessary conditions for a point to be efficient or weak efficient solution for general convex vector optimization problems, as well as for the nonconvex quadratic multiobjective optimization problem, are established.     

Girsanov's theorem in Hilbert space and an application to the statistics of Hilbert space- valued stochastic differential equations

We prove Girsanov-type theorems for Hilbert space-valued stochastic differential equations and apply them to a parameter estimation problem for linear infinite dimensional stochastic differential equations. In particular we construct the asymptotic statistical theory of the estimator, proving strong consistency and asymptotic normality.