Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half‐space with a cavity C. Zero normal derivative is assumed at the boundary of the half‐space; differently, at ?C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ?C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half‐space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ?C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd.     

Vertical oscillations of a circular die on an elastic half-space with a cylindrical cavity

The problem on vertical oscillations of a circular die with a plane base that freely lies on an elastic half-space and contains a cylindrical cavity, whose axis is perpendicular to the plane of the base of the die and passes through its center, is considered. The problem is formulated in the form of paired integral equations that are related to integral Weber transforms. The paired equations are reduced to an equivalent Fredholm equation of the second kind. Some results of numerical calculations of amplitude-frequency characteristics of oscillations of the die are given. A comparative analysis with a known solution of an analogous problem for a continuous elastic half-space is given.Translated from Dinamicheskie Sistemy, No. 8, pp. 30–36, 1989.     

A parallel multisplitting solution of the least squares problem

The linear least squares problem, min_x∥Ax − b∥₂, is solved by applying a multisplitting (MS) strategy in which the system matrix is decomposed by columns into p blocks. The b and x vectors are partitioned consistently with the matrix decomposition. The global least squares problem is then replaced by a sequence of local least squares problems which can be solved in parallel by MS. In MS the solutions to the local problems are recombined using weighting matrices to pick out the appropriate components of each subproblem solution. A new two-stage algorithm which optimizes the global update each iteration is also given. For this algorithm the updates are obtained by finding the optimal update with respect to the weights of the recombination. For the least squares problem presented, the global update optimization can also be formulated as a least squares problem of dimension p. Theoretical results are presented which prove the convergence of the iterations. Numerical results which detail the iteration behavior relative to subproblem size, convergence criteria and recombination techniques are given. The two-stage MS strategy is shown to be effective for near-separable problems. © 1998 John Wiley & Sons, Ltd.     

Forced oscillations with a sliding regime range of a two-mass system interacting with a fixed stop : PMM vol. 37, n6, 1973, pp. 999–1006

We investigate, by the method developed in [1]. the forced oscillations with a sliding regime range of a two-mass system with elastic connection between the elements, impacting a fixed stop. The system being considered is a dynamic model for a number of vibrational mechanisms. Forced oscillations with a sliding regime range of a system with shock interactions are periodic motions accompanied by a period of an infinite succession of instantaneous collisions of two fixed elements of the model [2]. Within the framework of conditions of roughness of the parameter space [3], in this paper we study by the method of [1] periodic motions with a sliding regime range of a two-mass system with a stop. This problem was posed because in real systems the velocity recovery factor R changes from shock to shock, mainly taking small values (0, 0.2). At the same time, the regions of realizability of one-impact oscillations, in practice the most essential ones among motions with a finite number of interactions over a period, narrow down sharply as R decreases and becomes very small even for R < 0.6 [4]. Thus, the stability of the given operation can be ensured by a law of motion which is independent or weakly dependent on R (^*) (see footnote on the next page). By virtue of what has been said above, finite-impact periodic modes are little suitable for this purpose. Regions, delineated in the parameter space of the model being considered, of existence of stable periodic motions with a sliding regime range have proved to be sufficiently broad. By virtue of the adopted approximation of the sliding regime, the dynamic characteristics of these motions do not depend upon R. The circumstances mentioned confirm the practical value of motions with a sliding regime range in dynamic systems with impact interactions.     

Special John-Nirenberg-Campanato spaces via congruent cubes

Let p ∈ [1, ∞), q ∈ [1, ∞), α∈ R, and s be a non-negative integer. Inspired by the space JNp introduced by John and Nirenberg(1961) and the space B introduced by Bourgain et al.(2015), we introduce a special John-Nirenberg-Campanato space JN^con_(p,q,s) over Rⁿ or a given cube of R;with finite side length via congruent subcubes, which are of some amalgam features. The limit space of such spaces as p →∞ is just the Campanato space which coincides with the space BMO(the space of functions with bounded mean oscillations)when α = 0. Moreover, a vanishing subspace of this new space is introduced, and its equivalent characterization is established as well, which is a counterpart of the known characterization for the classical space VMO(the space of functions with vanishing mean oscillations) over Rⁿ or a given cube of Rⁿ with finite side length.Furthermore, some VMO-H¹-BMO-type results for this new space are also obtained, which are based on the aforementioned vanishing subspaces and the Hardy-type space defined via congruent cubes in this article. The geometrical properties of both the Euclidean space via its dyadic system and congruent cubes play a key role in the proofs of all these results.     

Optimal design of the cooling plunger cavity

An axisymmetric system of mould, glass piece, plunger and plunger cavity is considered. The state problem is given as a stationary head conduction process. The system includes the glass piece representing the heat source and is cooled inside the plunger cavity by flowing water and outside by the environment of the mould. The design variable is taken to be the shape of the inner surface of the plunger cavity. The cost functional is the second power of the norm in the weighted space L _r ² of difference of trace of temperature from given constant, which is evaluated on the outward boundary of the plunger. Existence and uniqueness of the state problem solution and existence of a solution of the optimization problem are proved.     

On the Stokes and Navier-Stokes System for Domains with Noncompact Boundary in Lq-spaces

We consider the L^q-theory of weak solutions of the Stokes and Navier-Stokes equations in two classes of unbounded domains with noncompact boundary, namely in perturbed half spaces which are obtained by a perturbation of the half space IRⁿ, and in aperture domains consisting of two disjoint half spaces separated by a wall but connected by a hole (aperture) through this wall. The proofs rest on the cut-off procedure and a new multiplier approach to the half space problem. In an aperture domain we additionally prescribe either the flux through the wall or the pressure drop at infinity to single out a unique solution. The nonlinear problem is solved for sufficiently small data and requires q =n/2, n ≥ 3, to estimate the nonlinearity.     

Torsional oscillations of a circular stamp on an elastic layer with a cylindrical cavity

The problem of torsional oscillations of a stamp that is linked with an elastic stratum which contains a cylindrical cavity is considered. The problem is formulated in the form of conjugate integral equations that are related to the integral Weber transforms. The conjugate equations are reduced to an equivalent Fredholm equation of the second kind.Translated from Dinamicheskie Sistemy, No. 9, pp. 54–59, 1990.     

On linear vibrational systems with one dimensional damping II

We are interested in the quadratic eigenvalue problem of damped oscillations where the damping matrix has dimension one. This describes systems with one point damper. A generic example is a linearn-mass oscillator fixed on one end and damped on the other end. We prove that in this case the system parameters (mass and spring constants) are uniquely (up to a multiplicative constant) determined by any given set of the eigenvalues in the left half plane. We also design an effective construction of the system parameters from the spectral data. We next propose an efficient method for solving the Ljapunov equation generated by arbitrary stiffness and mass matrices and a one dimensional damping matrix. The method is particularly efficient if the Ljapunov equation has to be solved many times where only the damping dyadic is varied. In particular, the method finds an optimal position of a damper in some 60n ³ operations. We apply this method to our generic example and show, at least numerically, that the damping is optimal (in the sense that the solution of a corresponding Ljapunov equation has a minimal trace) if all eigenvalues are brought together. We include some perturbation results concerning the damping factor as the varying parameter. The results are hoped to be of some help in studying damping matrices of the rank much smaller than the dimension of the problem.     

A construction of unitary linear systems

A construction is made of a unitary linear system whose transfer function is a given power seriesB(z) with operator coefficients such that multiplication byB(z) is an everywhere defined transformation in the space of square summable power series with vector coefficients. A condition is also given for the existence of an observable linear system with such a transfer function. For both constructions properties of the spaces are given which imply essential uniqueness of linear systems with given transfer functions. A canonical conjugate-isometric linear system is uniquely determined by its transfer function whenever the state space is a Pontryagin space.     

Second Order Elliptic Differential-Operator Equations with Unbounded Operator Boundary Conditions in <Emphasis Type="Italic">UMD</Emphasis> Banach Spaces

In a UMD Banach space E, we consider a boundary value problem for a second order elliptic differential-operator equation with a spectral parameter when one of the boundary conditions, in the principal part, contains a linear unbounded operator in E. A theorem on an isomorphism is proved and an appropriate estimate of the solution with respect to the space variable and the spectral parameter is obtained. In this way, Fredholm property of the problem is shown. Moreover, discreteness of the spectrum and completeness of a system of root functions corresponding to the homogeneous problem are established. Finally, applications of obtained abstract results to nonlocal boundary value problems for elliptic differential equations with a parameter in non-smooth domains are given.     

Analysis of a one-dimensional free boundary flow problem

A one-dimensional free surface problem is considered. It consists in Burgers’ equation with an additional diffusion term on a moving interval. The well-posedness of the problem is investigated and existence and uniqueness results are obtained locally in time. A semi-discretization in space with a piecewise linear finite element method is considered. A priori and a posteriori error estimates are given for the semi-discretization in space. A time splitting scheme allows to obtain numerical results in agreement with the theoretical investigations.Supported by the Swiss National Science Foundation     

Abstract Hankel Operators in Kreĭn Spaces

Hankel operators and their symbols, as generalized by V. Pták and P. Vrbová, are considered in the Kreĭn space setting. Under a generic assumption, without which the Krein space case may be untreatable, a necessary and sufficient condition for the existence of Hankel symbols for a given Hankel operator X is given. A parametric labeling of the Hankel symbols of X by means of Schur class functions is obtained. The proof is established by associating to the data of the problem an isometry V acting on a Kreĭn space so that there is a bijective correspondence between the symbols of X and the minimal unitary Hilbert space extensions of V . The result includes uniqueness criteria and a Schur like formula.     

A fourth‐order H1‐Galerkin mixed finite element method for Kuramoto–Sivashinsky equation

A H¹‐Galerkin mixed finite element method is applied to the Kuramoto–Sivashinsky equation by using a splitting technique, which results in a coupled system. The method described in this article may also be considered as a Petrov–Galerkin method with cubic spline space as trial space and piecewise linear space as test space, since the second derivative of a cubic spline is a linear spline. Optimal‐order error estimates are obtained without any restriction on the mesh for both semi‐discrete and fully discrete schemes. The advantage of this method over that presented in Manickam et al., Comput. Math. Appl. vol. 35(6) (1998) pp. 5–25; for the same problem is that the size (i.e., (n + 1) × (n + 1)) of each resulting linear system is less than half of the size of the linear system of the earlier method, where n is the number of subintervals in the partition. Further, there is a requirement of less regularity on exact solution in this method. The results are validated with numerical examples. Finally, instability behavior of the solution is numerically captured with this method.     

Antiplane deformation of an isotropic massif with a curvilinear cavity

We solve the problem of antiplane deformation for an infinite isotropic massif with a curvilinear cavity in which a harmonic shear wave is propagating. The solution of the problem, which is based on the application of the theory of functions of a complex variable, is reduced to finding unknown constants from a system of linear algebraic equations. Numerical results are given for a circular cavity. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 115–119.     

Asymptotics of small spacings for subsonic nonstationary streamlined flow of a thin airfoil close to a solid boundary

In an asymptotic approximation of small spacings an analytic solution to the problem on harmonic oscillations of a thin airfoil which is moving with a subsonic velocity near a solid plane boundary is given. Results of a computation of the lifting force are given.Translated from Dinamicheskie Sistemy, No. 7, pp. 48–53, 1988.     

BOUNDARY LAYER SOLUTION FOR p-SYSTEM WITH ARTIFICIAL VISCOSITY

An initial-boundary values problem in the half space （0, ∞ ） for p-system with artificial viscosity is investigated. It is shown that there exists a boundary layer solution. It is further proved that the boundary layer solution is nonlinear stable with arbitrarily large perturbation. The proof is given by an elementary energy method.     

Synthesis of controllers for damping oscillations in dynamical systems under uncertainty

We review the modern approaches to the synthesis of robust H _∞ controllers that ensure optimal damping of oscillations in dynamical systems under uncertainty. In the synthesis method based on Riccati equations, these many-parameter equations can be solved only when the parameters are contained in a bounded parallelepiped with given boundaries. The synthesis of a robust H _∞ output control for systems with unknown bounded parameters is reducible to the solution of an optimization problem constrained by a system of linear matrix inequalities. The proposed controller synthesis algorithms are implemented using standard MATLAB procedures. The efficiency of the proposed methods and algorithms is demonstrated in application to optimal damping of oscillations in a parametrically excited pendulum. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 87–104, 2004.     

The inverse scattering problem for cavities with impedance boundary condition

We consider the inverse scattering problem of determining the shape of a cavity with impedance boundary condition from sources and measurements placed on a curve inside the cavity. It is shown that both the shape ?D\partial D of the cavity and the surface impedance λ are uniquely determined by the measured data and numerical methods are given for determining both ?D\partial D and λ where neither one is known a priori. Numerical examples are given showing the viability of our method.     

Evolution Differential Inclusion with Projection for Solving Constrained Nonsmooth Convex Optimization in Hilbert Space

This paper introduces a projection subgradient system modeled by an evolution differential inclusion to solve a class of hierarchical optimization problems in Hilbert space. Basing on the Moreau–Yosida approximation, we prove the global existence and uniqueness of the solution of the proposed evolution differential inclusion with projection and the unique solution of the proposed system is just its “slow solution” when the constrained set is defined by the affine equalities. When the outer layer objective function ψ is strongly convex, any solution of the proposed system is strongly convergent to the unique minimizer of the constrained optimization problem, while, the strongly convergence is also given when the inner layer objective function ϕ is strongly convex. Furthermore, we present some other optimization problem models, which can be solved by the proposed system. All the results obtained are new not only in the infinite dimensional Hilbert space framework but also in the finite dimensional space.