On a variational inequality associated with a stopping game combined with a control

We study a non-linear elliptic variational inequality which corresponds to a zero-sum stopping game (Dynkin game) combined with a control. Our result is a generalization of the existing works by Bensoussan [ Stochastic Control by Functional Analysis Methods (North-Holland, Amsterdam), 1982], Bensoussan and Lions [ Applications des Inéquations Variationnelles en Contrôle Stochastique (Dunod, Paris), 1978] and Friedman [ Stochastic Differential Equations and Applications (Academic Press, New York), 1976] in the sense that a non-linear term appears in the variational inequality, or equivalently, that the underlying process for the corresponding stopping game is subject to a control. By using the dynamic programming principle and the method of penalization, we show the existence and uniqueness of a viscosity solution of the variational inequality and describe it as the value function of the corresponding combined-stochastic game problem.     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

The motion of a body with a cavity partly filled with a viscous liquid

Many papers are concerned with the dynamics of a rigid body with a cavity filled with liquid (see the bibliography in [1]). The present paper deals with the motion of a rigid body having a cavity partly filled with a viscous incompressible liquid, and having a free surface. The shape of the cavity is arbitrary. The problem is considered in a linear formulation. The oscillations of the body with respect to its center of inertia and the motion of the liquid in the cavity are assumed small. The viscosity of the liquid is considered low. The solution of the problem of the oscillations of a body with a cavity partly filled with an ideal liquid is used as an initial approximation [1 to 6]. The viscosity is taken into consideration by the boundary layer method used before in similar problems [1 and 7 to 10). General equations are derived for the dynamics of a body filled with a liquid, for an arbitrary form of cavity. The coefficients of those integro-differential equations depend only on the solution of the problem of the oscillations of a body with a cavity of the given form filled with an ideal liquid. Since the corresponding problem has been solved for cavities of many forms [1 to 6, 11 and 12] in the case of an ideal liquid, the determination of the characteristic coefficients is reduced to the evaluation of quadratures. Several particular cases of motion are considered.     

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.     

Placing a finite size facility with a center objective on a rectangular plane with barriers

This paper addresses the finite size 1-center placement problem on a rectangular plane in the presence of barriers. Barriers are regions in which both facility location and travel through are prohibited. The feasible region for facility placement is subdivided into cells along the lines of Larson and Sadiq [R.C. Larson, G. Sadiq, Facility locations with the Manhattan metric in the presence of barriers to travel, Operations Research 31 (4) (1983) 652–669]. To overcome complications induced by the center (minimax) objective, we analyze the resultant cells based on the cell corners. We study the problem when the facility orientation is known a priori. We obtain domination results when the facility is fully contained inside 1, 2 and 3-cornered cells. For full containment in a 4-cornered cell, we formulate the problem as a linear program. However, when the facility intersects gridlines, analytical representation of the distance functions becomes challenging. We study the difficulties of this case and formulate our problem as a linear or nonlinear program, depending on whether the feasible region is convex or nonconvex. An analysis of the solution complexity is presented along with an illustrative numerical example.     

Sets with a mode

LetM be a point andS be a compact set inR ² such thatS is the closure of its interior. The theorem desired says that ifM is a mode ofS thenS is convex and centrally symmetric with respect toM. Some conditions on the boundary ofS are needed for the proof given.     

Sampling with a String

Kramer's sampling theorem forms a bridge between the Whittaker-Shannon-Kotel'nikov sampling theorem and boundary-value problems. It has been shown that sampling expansions associated with Sturm-Liouville boundary-value problems are Lagrange-type sampling series, i.e., Lagrange series with infinitely many terms converging to entire functions. String theory as developed by Feller, Kac, and Krein, is a generalization of the Sturm-Liouville theory. We investigate sampling series associated with strings and compare them with those associated with Sturm-Liouville problems. We show that unlike sampling series associated with Sturm-Liouville problems, those associated with strings include not only Lagrange-type sampling series, but also Lagrange polynomial interpolation.     

Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert

Two problems of the interaction of a hollow circular cylinder with load-free ends and an unbounded plate with a cylindrical cavity and a symmetrically imbedded rigid insert are considered. Homogeneous solutions are found and the generalized orthogonality of these solutions is used when the modified boundary conditions are satisfied. As a result, we have a system of two integral equations in functions of the displacements of the outer and inner surfaces of the hollow cylinder. These functions are sought in the form of sums of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of linear algebraic equations obtained are regularized by the introduction of small positive parameters. Since the elements of the matrices of the systems as well as the contact stresses are defined by poorly converging numerical and functional series, an efficient method for calculating of the remainders of the above-mentioned series is developed. Formulae are found for the contact pressure distribution function and the integral characteristic. Examples of the calculation of the interaction of the cylinder and the plate with an insert are given.The method of solving contact problems described here has been used earlier1, 2 and the generalized orthogonality of the solutions found for bodies of finite dimensions, that is, for a rectangle and cylinders of finite length, is its basis. Problems for hollow cylinders with a band ² and an insert reduce to a system of two integral equations, and the problem for a rectangle¹ reduces to one integral equation. Solving these integral equations, ill-posed systems of linear algebraic equations are obtained which are subject to regularization³.     

Calculating a function of a matrix with a real spectrum

Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f(T) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries t_ii are arranged in increasing order. To avoid calculations using the differences t_ii ? t_jj with close (including equal) t_ii and t_jj, it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f(T) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits.

     

Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides

The plane contact problem of the indentation of a rigid punch into a base-sucured elastic rectangle with stress-free sides is considered. The problem is solved by a method tested earlier and reduces to a system of two integral equations in functions describing the displacement of the surface of the rectangle outside the punch and the normal or shear stress on its base. These functions are sought in the form of the sum of trigonometric series and an exponential function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result of this are regularized by introducing small positive parameters. Because the matrix elements of the systems, and also the contact stresses, are defined by poorly converging numerical and functional series, the previously developed method of summation of these series is used. The contact pressure distribution and the dimensionless indenting force are found. Examples of a plane punch calculation are given.     

The direct product of a hopfian group with a group with cyclic ccnter

In this paper we continue our study of hopficity begun in [1], [2], [3], [4] and [5]. LetA be hopfian and letB have a cyclic center of prime power order. We improve Theorem 4 of [2] by showing that ifB has finitely many normal subgroups which form a chain (we sayB isn-normal), thenAxB is hopfian. We then consider the case whenB is ap-group of nilpotency class 2 and show that in certain casesAxB is hopfian.     

The direct product of a hopfian group with a group with cyclic ccnter

In this paper we continue our study of hopficity begun in [1], [2], [3], [4] and [5]. LetA be hopfian and letB have a cyclic center of prime power order. We improve Theorem 4 of [2] by showing that ifB has finitely many normal subgroups which form a chain (we sayB isn-normal), thenAxB is hopfian. We then consider the case whenB is ap-group of nilpotency class 2 and show that in certain casesAxB is hopfian.     

Homogenization of the heat equation for a domain with a network of pipes with a well-mixed fluid

We consider the linear heat equation in a domain occupied by a solid material with a network of pipes in which a well-mixed fluid is circulating. The temperature of the fluid in the pipe is uniform and its time variation is determined by the thermal flux on the wall of the pipe, plus a given internal source; continuity of the temperature across the pipe is also assumed. We suppose that we deal with a periodic geometry, with cells of size with inclusions of size r_g; we study in detail in the case r, referring to a previous paper for the case r In the limit »0 we get a homogenized equation. The limit depends strongly on the ratio between the time variation of the temperature in the inclusions and the thermal flux through the interface. The homogenized equation has a new specific heat, which depends on the porosity and the constant of proportionality between the time variation of temperature and the flux on the boundary of the pipe. We also have a new thermal conductivity depending on the microstructure, and volume sources appear. The main tool is the energy method and we generalize the classical results for the more standard boundary conditions for parabolic equations. Finally, we consider the network of pipes forming a random ball structure. We prove convergence for this case. The homogenized equation is of the same form as in the periodic case but auxiliary problems are stochastic.     

Boundary value problems of electroelasticity for a plate with an inclusion and a half-space with a slit

Problems of determining the mechanical and electrical fields in a piezoelectric plate reinforced with an inclusion or in a half-space weakened by a cut are considered. Using the methods of the theory of analytic functions these problems are reduced to a system of singular integro-differential equations (for a plate) or to a singular integral equation with a fixed singularity (for a half-space). Approximate and exact solutions of the problems are obtained by the method of orthogonal polynomials and integral transforms.