Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

In this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non-homogeneous obstacles using convolution quadrature (CQ) techniques for the time variable and coupled boundary element method/finite element method for the space variable. The applicability of CQ to waves requires polynomial type bounds for operators related to the operator Δ − s ² in the right half complex plane. We propose a new systematic way of dealing with this problem, both at the continuous and semidiscrete-in-space cases. We apply the technique to three different situations: scattering by a group of sound-soft and -hard obstacles, by homogeneous and non-homogeneous obstacles.     

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

Summary  We consider the numerical treatment of second kind integral equations on the real line of the form

A hybrid method for computing Lyapunov exponents

In this paper we propose a numerical method for computing all Lyapunov coefficients of a discrete time dynamical system by spatial integration. The method extends an approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223–237, 1999) who use a box approximation of an underlying ergodic measure and compute the first Lyapunov exponent from a spatial average of the norms of the Jacobian for the iterated map. In the hybrid method proposed here, we combine this approach with classical QR-oriented methods by integrating suitable R-factors with respect to the invariant measure. In this way we obtain approximate values for all Lyapunov exponents. Assuming somewhat stronger conditions than those of Oseledec’ multiplicative theorem, these values satisfy an error expansion that allows to accelerate convergence through extrapolation. W.-J. Beyn and A. Lust was supported by CRC 701 ‘Spectral Analysis and Topological Methods in Mathematics’. The paper is mainly based on the PhD thesis [27] of A. Lust.     

A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some appropriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.     

Positive solutions for second-order superlinear repulsive singular Neumann boundary value problems

In this paper we establish the multiplicity of positive solutions to second-order superlinear repulsive singular Neumann boundary value problems. It is proved that such a problem has at least two positive solutions under reasonable conditions. Our nonlinearity may be repulsive singular in its dependent variable and superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.      

A Feasibility Pump for mixed integer nonlinear programs

We present an algorithm for finding a feasible solution to a convex mixed integer nonlinear program. This algorithm, called Feasibility Pump, alternates between solving nonlinear programs and mixed integer linear programs. We also discuss how the algorithm can be iterated so as to improve the first solution it finds, as well as its integration within an outer approximation scheme. We report computational results. P. Bonami is supported in part by a grant from IBM and by ANR grant BLAN06-1-138894. G. Cornuéjols is supported in part by NSF grant CMMI-0653419, ANR grant BLAN06-1-138894 and ONR grant N00014-03-1-0188. Part of this research was carried out when Andrea Lodi was Herman Goldstine Fellow of the IBM T.J. Watson Research Center whose support is gratefully acknowledged. F. Margot is supported in part by a grant from IBM and by ONR grant N00014-03-1-0188.     

On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations

We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness of such solutions for the first initial boundary value problem. Dedicated to Professor Felix Browder     

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.     

A priori estimates for the vector p-Laplacian with potential boundary conditions

This paper deals with the solvability of the boundary value problem

A colorful theorem on transversal lines to plane convex sets

We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the plane has the property that any three differently colored members have a transversal line that meet the sets consistently with the numbering, then there exists a color such that all the convex sets of that color have a transversal line. All authors are partially supported by CONACYT research grant 5040017.     

Stabilized boundary element methods for exterior Helmholtz problems

In this paper we describe and analyze some modified boundary element methods to solve the exterior Dirichlet boundary value problem for the Helmholtz equation. As in classical combined field integral equations also the proposed approach avoids spurious modes. Moreover, the stability of related modified boundary element methods can be shown even in the case of Lipschitz boundaries. The proposed regularization is done based on boundary integral operators which are already included in standard boundary element formulations. Numerical examples are given to compare the proposed approach with other already existing regularized formulations.     

Thin boundary layers of chiral smectics

We consider a reduced Landau–de Gennes energy functional which describes a chiral smectic liquid crystal with large elastic coefficients. We prove that, according to this model, chiral smectics exhibit behavior which is similar to surface superconductivity: a thin layer of smectics near the boundary, and cholesterics in the bulk of the material. We obtain this behavior for a wide region in the parameter space. We show that in a certain limit case this boundary layer can determine the direction of the helical axis of the cholesterics.     

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation on a closed manifold M, it is known that is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form (relatively to a suitable riemannian metric μ) is zero (cf. álvarez in Ann Global Anal Geom 10:179–194, 1992). In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group , where (cf. Masa in Comment Math Helv 67:17–27, 1992). By the Poincaré Duality (cf. Kamber et and Tondeur in Astérisque 18:458–471, 1984) this last condition is equivalent to the non-vanishing of the basic twisted cohomology group , when M is oriented. When M is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF). J. I. Royo Prieto was partially supported by EHU06/05, by a PostGrant from the Basque Government and by the MCyT of the Spanish Government. R. Wolak was partially supported by the KBN grant 2PO3A 021 25.     

The existence of three positive solutions for some nonlinear boundary value problems on the half-line

In this paper, by introducing a new operator, improving and generating a p-Laplace operator for some p > 1, we consider the existence of triple positive solutions for some nonlinear m-point boundary value problems on the half-line

Scattering of an incident acoustic wave by an elastic sphere in a shallow water

The scattering of acoustic waves by an elastic sphere in a shallow ocean wave guide is investigated taking into account the shear waves which can exist in addition to compressional waves in scatterers of solid material. Expressions for the scattered waves are given. Numerical values for a quantity called the farfield form function for various depth are presented in graphical forms.      

A unified theory for modifications of<Emphasis Type="Italic">g</Emphasis>-closed sets

We introduce a new set calledmg-closed which is defined on a family of sets satisfying some minimal conditions. This set enables us to unify certain kind of modifications of generalized closed sets due to Levine [17].     

Convergence analysis of sectional methods for solving breakage population balance equations-II: the cell average technique

This work presents the convergence of the cell average technique (Kumar et al. in Powder Technol 179:205–228, 2007) for solving breakage population balance equation. Similarly to our paper Kumar and Warnecke (Numerische Math, 2008) of this series, we study convergence on four different types of meshes. A second order convergence is proved for uniform, locally uniform and non-uniform smooth meshes. Finally the scheme is analyzed on random mesh and it is found that the scheme is only first order accurate. Nevertheless we obtain for locally uniform as well as for random mesh one order higher accuracy than the fixed pivot technique discussed by the authors in the first paper. All mathematical observations of convergence analysis are also validated numerically and numerical results are compared with the results of the first part.     

Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations , u(0) = u₀, u_t (0) = v₀, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation , u(0) = u₀, endowed with Dirichlet boundary conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

A remark on roots of polynomials with positive coefficients

A short proof of a theorem of Dubickas on roots of polynomials with positive rational coefficients is presented.