Asymptotics of a solution of a nonlinear system of diffusion of a magnetic field into a substance

We study the asymptotic behavior at large time of a solution to a system of nonlinear integro-differential equations which arises in mathematical modeling of diffusion of a magnetic field into a substance. We establish the corresponding stabilization rate.     

Canonical frame of a curve on a conformal plane

We show how differential geometry of smooth curves on the conformal plane can be studied by Élie Cartan’s method of exterior forms and moving frames. We find the canonical form of the derivation equations of a curve (which is not a circle) in the case of a semi-isotropic frame. We give a new proof of the theorem that states that curves of constant (in particular, zero) conformal curvature are loxodromes. We integrate the system of structure equations of the isotropy subgroup of a point.     

Essential spectrum of a model operator associated with a three-particle system on a lattice

We consider a model operator H associated with the system of three particles interacting via nonlocal pair potentials on a ν-dimensional lattice. We identify channel operators and use their spectra to describe the position and structure of the essential spectrum of H. We obtain an analogue of the Faddeev equation for the eigenfunctions of H.     

On a classical problem with a complex-valued coefficient and the spectral parameter in a boundary condition

We consider a classical problem that arises when studying natural vibrations of a loaded string. We assume that the coefficient playing the role of a physical parameter can take complex values. We discuss the completeness, minimality, and basis property of the system of root functions.     

Spectral properties of a Hamiltonian of a four-particle system on a lattice

We consider the Hamiltonian of a system of four arbitrary quantum particles with two-particle contact (noncompact) interaction potentials on a three-dimensional lattice perturbed by three-particle contact potentials. We describe the location of the essential spectrum of the Schrödinger operator corresponding to a four-particle system.     

Interaction of a breather with a magnetization wave in a ferromagnet with light-axis anisotropy

We use the dressing method to find exact solutions of the Landau-Lifshitz equation for a ferromagnet with light-axis anisotropy. These solutions describe the interaction of a nonlinear precession wave of arbitrary amplitude with solitons. We analyze the change of the internal structure and the physical parameters of the solitons as a result of their interaction with the magnetization wave. We find an infinite series of integrals of motion that stabilize the soliton on the background of the pumping wave.     

Solvability of a dynamic contact problem between a piezoelectric body and a conductive foundation

We consider a mathematical model which describes the dynamic process of contact between a piezoelectric body and an electrically conductive foundation. We model the material’s behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the normal compliance condition and a regularized electrical conductivity condition. We derive a variational formulation for the problem and then, under a smallness assumption on the data, we prove the existence of a unique weak solution to the model. We also investigate the behavior of the solution with respect the electric data on the contact surface and prove a continuous dependence result. Then, we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We treat the contact by using a penalized approach and a version of Newton’s method. We implement this scheme in a numerical code and, in order to verify its accuracy, we present numerical simulations in the study of two-dimensional test problems. These simulations provide a numerical validation of our continuous dependence result and illustrate the effects of the conductivity of the foundation, as well.     

Study of a discrete dynamic system in a neighborhood of a quasi-periodic trajectory

We study a discrete system in a neighborhood of a quasi-periodic trajectory. We obtain conditions for reducing a system in this neighborhood to a system with quasi-periodic coefficients. We determine the behavior of this system under the action of small perturbations.This work was prepared with the financial support of the Ukrainian State Committee on Science and Technology.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1702–1711, December, 1992.     

Shadowing in a neighborhood of a separatrix

We consider a model example of a diffeomorphism of a two-dimensional manifold with two hyperbolic fixed points joined by a separatrix. We show that, in a tubular neighborhood of the separatrix, every pseudotrajectory satisfying an additional error smallness condition is shadowed by an exact trajectory.     

A surface containing a line and a circle through each point is a quadric

We prove that a surface in 3-dimensional Euclidean space containing a line and a circle through each point is a quadric. We also give some particular results on the classification of surfaces containing several circles through each point.     

Weak solutions of a mixed problem in a half-strip for a generalized Kawahara equation

We consider a mixed problem in a half-strip for a generalized Kawahara equation describing long nonlinear waves in inhomogeneous media with weak dispersion. We study the existence and uniqueness of generalized solutions in Sobolev weight spaces.     

Numerical analysis of a model for phase separation of a multicomponent alloy

We consider a fully discrete implicit finite-element approximationof a model for the phase separation of a multi-component alloy.We prove existence, uniqueness and stability of the numericalsolution for a sufficiently small time step. We prove convergenceto the solution of the associated continuous problem. We performa linear stability analysis of the equation and describe somenumerical experiments.     

Investigation of a nonlinear difference equation in a Banach space in a neighborhood of a quasiperiodic solution

We investigate the behavior of a diserete dynamical system in a neighborhood of a quasiperiodic trajeetory for the case of an infinite-dimensional Banach space We find conditions sufficient for the system considered to reduce, in such a neighborhood, to a system with quasiperiodic coefficients. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1661–1676, December, 1997.     

Resolution of a scheduling problem in a flowshop robotic cell

We develop in this paper a generic and precise identification of a scheduling problem in a flexible manufacturing system. We consider a flowshop robotic cell that processes several jobs. We assume that there is no intermediate buffer between machines. So, jobs may be blocked when downstream machines are busy. We present an integer programming model to determine the sequence of jobs that minimizes the makespan criterion. In order to solve large size problems, we propose a genetic algorithm (GA). Finally, computational experiments are proposed in order to compare the makespan returned by the GA to a lower bound.     

Reinforcement of a thin plate by a thin layer

We study the bending of a thin plate, stiffened with a thin elastic layer, of thickness δ. We describe the complete construction of an asymptotic expansion with respect to δ of the solution of the Kirchhoff–Love model and give optimal estimates for the remainder. We identify approximate boundary conditions, which take into account the effect of the stiffener at various orders. Thanks to the tools of multi‐scale analysis, we give optimal estimates for the error between the approximate problems and the original one. We deal with a layer of constant stiffness, as well as with a stiffness in δ^?1. Copyright © 2007 John Wiley & Sons, Ltd.     

Sufficient Conditions for a Minimum of a Strongly Quasiconvex Function on a Weakly Convex Set

Mathematical Notes - We consider a finite-dimensional minimization problem for a strongly quasiconvex function on a weakly convex set. We obtain sufficient conditions for its solution expressed in...     

Extending a matroid by a cocircuit

Our main result describes how to extend a matroid so that its ground set is a modular hyperplane of the larger matroid. This result yields a new way to view Dowling lattices and new results about line-closed geometries. We complement these topics by showing that line-closure gives simple geometric proofs of the (mostly known) basic results about Dowling lattices. We pursue the topic of line-closure further by showing how to construct some line-closed geometries that are not supersolvable.     

Optimizing a Particular Real Root of a Polynomial by a Special Cylindrical Algebraic Decomposition

We study the problem of optimizing over parameters a particular real root of a polynomial with parametric coefficients. We propose an efficient symbolic method for solving the optimization problem based on a special cylindrical algebraic decomposition algorithm, which asks for a semi-algebraic decomposition into cells in terms of number-of-roots-invariance.     

Exponentially damped operators for a harmonic oscillator linearly coupled to a heat bath

We consider the problem of the dissipative dynamics of a harmonic oscillator linearly coupled to a heat bath. We demonstrate that in addition to the mean energy, there exists an infinite series of quantities exponentially decreasing in time that are means of polynomials of the system Hamiltonian. We obtain the spectrum of the corresponding relaxation times. We propose a method for representing the time characteristics of the system in terms of operators corresponding to the exponentially damped observables. We obtain a recurrence relation for these operators.     

Generalized Kazdan-Warner equations associated with a linear action of a torus on a complex vector space

Geometriae Dedicata - We introduce generalized Kazdan-Warner equations on Riemannian manifolds associated with a linear action of a torus on a complex vector space. We show the existence and the...