An application of the limiting absorption principle to a mixed boundary value problem in an infinite strip

We present a complete proof of the existence and uniqueness of solutions of a mixed boundary value problem for the homogeneous Laplace equation in an unbounded parallel strip based on the principle of limiting absorption.     

The limiting absorption principle in the problem of a transparent wedge

The problem of diffraction of a plane wave by a transparent wedge is considered (the wave numbers inside the wedge and outside it are different). Absorption in a medium is assumed to occur. The existence of a solution satisfying the limiting absorption principle is proved. The existence theorem and the fact that the limit passage can be performed are proved for the corresponding system of equations. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 138–148.     

A proof of the abstract limiting absorption principle by energy estimates

We give a proof in an abstract setting of various resolvent estimates including the limiting absorption principle starting from a Mourre positive commutator estimate, using standard energy estimates arguments.     

The limiting absorption principle for the two-dimensional inhomogeneous anisotropic elasticity system

In this work we establish the limiting absorption principle for the two-dimensional steady-state elasticity system in an inhomogeneous aniso- tropic medium. We then use the limiting absorption principle to prove the existence of a radiation solution to the exterior Dirichlet or Neumann boundary value problems for such a system. In order to define the radiation solution, we need to impose certain appropriate radiation conditions at infinity. It should be remarked that even though in this paper we assume that the medium is homogeneous outside of a large domain, it still preserves anisotropy. Thus the classical Kupradze's radiation conditions for the isotropic system are not suitable in our problem and new radiation conditions are required. The uniqueness of the radiation solution plays a key role in establishing the limiting absorption principle. To prove the uniqueness of the radiation solution, we make use of the unique continuation property, which was recently obtained by the authors. The study of this work is motivated by related inverse problems in the anisotropic elasticity system. The existence and uniqueness of the radiation solution are fundamental questions in the investigation of inverse problems.

     

A limiting absorption principle for scattering problems with unbounded obstacles

A generalized mode matching method that applies to a wide class of scattering problems is developed in the time harmonic two‐dimensional Helmholtz case. This method leads by variational means to an integro‐differential formulation whose unknown is the trace of the field on an unbounded one‐dimensional interface. The well‐posedness is proved after a careful study of the rather original functional framework. Owing to a fundamental density result—based upon some properties of a singular integral operator similar to the Hilbert transform—the limiting absorption principle related to this original formulation is established. Finally, two other situations are emphasized. Copyright © 2001 John Wiley & Sons, Ltd.     

An uncertainty principle on the group of rigid motions of the plane

Let M(2) be the group of rigid motions of the plane. The Fourier transform and the Plancherel formula on M(2) can be explicitly given by the general group representation theory. Using this fact, we establish a kind of uncertainty principle on M(2). The result can easily be generalized to higher dimensional cases. An application of the result yields an uncertainty principle on the Euclidean spaces obtained by R. S. Strichartz.     

An application of the fourier transform to sections of star bodies

We express the volume of central hyperplane sections of star bodies inR ⁿ in terms of the Fourier transform of a power of the radial function, and apply this result to confirm the conjecture of Meyer and Pajor on the minimal volume of central sections of the unit balls of the spacesℓ _p ⁿ with 0<p<2. Research supported in part by the NSF Grant DMS-9531594.     

Incomplete comparison principle in problems about surface waves trapped by fixed and freely floating bodies

We consider problems about interaction of surface waves with fixed and freely floating bodies. We propose a new boundary value problem with integro–differential boundary conditions. This problem has physical sense only in the case of a totally submerged body, but, in the case of symmetry in the geometric or physical sense, the nonemptiness of its discrete spectrum provides the existence of eigenvalues of both problems for a fixed body and a freely floating body (in the last case, under some additional requirements which make the comparison principle incomplete). Bibliography: 25 titles. Illustrations: 5 figures.     

Analysis of critical motions of floating structures

Validation of numerical methods for describing the motion of a ship in sea conditions by adequate experiments is a major research field in ocean engineering. For the development of a method for the systematic determination of critical and safe operational conditions and for the classification of capsize scenarios bifurcation analyses are performed. The computational effort for these analyses is enormous using a full model describing the nonlinear dynamics of a floating body. Therefore, a method for model reduction is currently being developed at the Institute of Mechanics and Ocean Engineering at TUHH. Bases for the validation of this new method are experiments conducted in the institute's wave tank. The determination of position and attitude of the body is performed with an integrated measurement system: An inertial measurement unit and a video system are combined using an extended Kalman Filter. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An invariance principle for semimartingale reflecting Brownian motions in an orthant

Semimartingale reflecting Brownian motions in an orthant (SRBMs) are of interest in applied probability because of their role as heavy traffic approximations for open queueing networks. It is shown in this paper that a process which satisfies the definition of an SRBM, except that small random perturbations in the defining conditions are allowed, is close in distribution to an SRBM. This perturbation result is called an invariance principle by analogy with the invariance principle of Stroock and Varadhan for diffusions with boundary conditions. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. In a subsequent paper, the invariance principle is used to give general conditions under which a heavy traffic limit theorem holds for open multiclass queueing networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

Large deflection analysis of floating roofs subjected to earthquake ground motions

Storage tanks with floating roofs have suffered severe damages during past earthquakes. To evaluate the seismic response of the cylindrical liquid storage tanks with floating roofs Hamilton’s variational principle is used. This study investigates the seismic response of the tanks accounting for nonlinearity due to large deflection of the deck plate. The ground motions include the long-period far-field record of Tokachi-oki, the near-source record of Kobe and far-field record of El Centro. It is found that accounting for large deflections in some cases could slightly magnify the roof deflection. However, usually the suppressing effect of the large deflection is more pronounced. Moreover, the frequency range in which large deflections have a suppressing effect is where ground motions are rich in the frequency content. These results are applicable for all of the ground motions considered in the analysis.     

An application of Neustadt's abstract maximum principle to probability kinematics

An important class of problems in philosophy can be formulated as mathematical programming problems in an infinite-dimensional vector space. One such problem is that of probability kinematics: the study of how an individual ought to adjust his degree-of-belief function in response to new information. Much work has recently been done to establish maximum principles for these generalized programming problems (Refs. 3–4). Perhaps, the most general treatment of the problem presented to date is that by Neustadt (Ref. 1). In this paper, the problem of probability kinematics is formulated as a generalized mathematical programming problem and necessary conditions for the optimal revised degree-of-belief function are derived from an abstract maximum principle contained in Neustadt's paper.This work was supported by the National Research Council of Canada.The author is grateful to G. J. Lastman and J. A. Baker of the University of Waterloo for numerous suggestions made for improvement of this paper. The problem of probability kinematics was brought to the author's attention by W. L. Harper of the University of Western Ontario.     

Optimal limiting absorption principle for a Schrödinger type operator on a Lipschitz cylinder

Spectral properties of a Schrödinger type operator on a cylinder is established by a version of Mourre's conjugate operator method. In particular, the limiting absorption principle is proved in an optimal scale of Besov spaces.     

The limiting angle of certain riemannian brownian motions

Let M be a Cartan-Hadamard manifold of dimension d ≧ 3, let p ? M and x = exp {r(x)θ(x)} be geodesic polar coordinates with pole p and let X be the Brownian motion on M. Let Sect_M(x) denote the sectional curvature of any plane section in M_x. We prove that for each c > 2, there is a 0 < β < 1 such that if - L²r(x)^2β ≦ Sect_M(x) ≦ -cr(x)^?2 for all x in the complement of a compact set, then lim_{t → ∞} θ(X_t) exists a.s. and defines a nontrivial invariant random variable. The Dirichlet problem at infinity and a conjecture of Greene and Wu are also discussed.     

An application of the smooth variational principle to the existence of nontrivial invariant subspaces

We show that the consideration of Gâteaux smooth functions on Banach spaces which admit an equivalent Gâteaux smooth norm allows us to show that certain linear operators have nontrivial closed invariant subspaces. It is in particular the case of all operators on a real Banach space which admit a moment sequence.     

Computation of flow-induced motion of floating bodies

A computational procedure for the prediction of motion of rigid bodies floating in viscous fluids and subjected to currents and waves is presented. The procedure is based on a coupled iterative solution of equations of motion of a rigid body with up to six degrees of freedom and the Reynolds-averaged Navier–Stokes equations describing the two- or three-dimensional fluid flow. The fluid flow is predicted using a commercial CFD package which can use moving grids made of arbitrary polyhedral cells and allows sliding interfaces between fixed and moving grid blocks. The computation of body motion is coupled to the CFD code via user-coding interfaces. The method is used to compute the 2D motion of floating bodies subjected to large waves and the results are compared to available experimental data, showing favorable agreement.