Distribution of poles for scattering on the real line

The density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential. In the case of a potential with finite singularities at the endpoints of the support, asymptotic formulae for the poles are given, while in the C₀^∞ case, an example of a potential with infinitely many scattering poles on iR is constructed. The scattering amplitude of a compactly supported potential is also characterized.     

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

A direct and inverse scattering theory on the full line is developed for a class of first-order selfadjoint 2n×2n systems of differential equations with integrable potential matrices. Various properties of the corresponding scattering matrices including unitarity and canonical Wiener-Hopf factorization are established. The Marchenko integral equations are derived and their unique solvability is proved. The unique recovery of the potential from the solutions of the Marchenko equations is shown. In the case of rational scattering matrices, state space methods are employed to construct the scattering matrix from a reflection coefficient and to recover the potential explicitly.Dedicated to Israel Gohberg on the Occasion of his 70^th Birthday     

On a method for solving the inverse scattering problem on the line

A method for solving the inverse scattering problem on the line is proposed. It is based on a Fourier‐Laguerre series representation of the integral transmutation kernel. Substitution of the representation into the Gel'fand‐Levitan‐Marchenko equation leads to a linear algebraic system of equations and consequently to a simple algorithm for recovering the potential.     

Exponential dichotomy on the real line: SVD and QR methods

In this work we show when and how techniques based on the singular value decomposition (SVD) and the QR decomposition of a fundamental matrix solution can be used to infer if a system enjoys—or not—exponential dichotomy on the whole real line.     

The no-response approach and its relation to non-iterative methods for the inverse scattering

This paper addresses the inverse obstacle scattering problem. In the recent years several non-iterative methods have been proposed to reconstruct obstacles (penetrable or impenetrable) from near or far field measurements. In the chronological order, we cite among others the linear sampling method, the factorization method, the probe method and the singular sources method. These methods use differently the measurements to detect the unknown obstacle and they require the use of many incident fields (i.e. the full or a part of the far field map). More recently, two other approaches have been added. They are the no-response test and the range test. Both of them use few incident fields to detect some informations about the scatterer. All the mentioned methods are based on building functions depending on some parameter. These functions share the property that their behaviors with respect to the parameter change drastically. The surface of the obstacle is located at most in the interface where these functions become large. The goal of this work is to investigate the relation between some of the non-iterative reconstruction schemes regarding the convergence issue. A given method is said to be convergent if it reconstructs a part or the entire obstacle by using few or many incident fields respectively. For simplicity we consider the obstacle reconstruction problem from far field data for the Helmholtz equation. Gen Nakamura is partially supported by Grant-in-Aid for Scientific research (B)(2)(N.14340038) of Japan Society for Promotion of Science. Mourad Sini is supported by Japan Society for Promotion of Science.     

An inverse eigenvalue problem: ComputingB-stable Runge-Kutta methods having real poles

The implementation of implicit Runge-Kutta methods requires the solution of large sets of nonlinear equations. It is known that on serial machines these costs can be reduced if the stability function of ans-stage method has only ans-fold real pole. Here these so-called singly-implicit Runge-Kutta methods (SIRKs) are constructed utilizing a recent result on eigenvalue assignment by state feedback and a new tridiagonalization, which preserves the entries required by theW-transformation. These two algorithms in conjunction with an unconstrained minimization allow the numerical treatment of a difficult inverse eigenvalue problem. In particular we compute an 8-stage SIRK which is of order 8 andB-stable. This solves a problem posed by Hairer and Wanner a decade ago. Furthermore, we finds-stageB-stable SIRKs (s=6,8) of orders, which are evenL-stable.     

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

Consider the reconstruction of Dirichlet-to-Neumann map(D-to-N map) from the far-field patterns of the scattered waves in inverse scattering problems, which is the first step in detecting the obstacle boundary by the probe method using far-field measurements corresponding to all incident plane waves. In principle, this problem can be reduced to solving an integral equation of the second kind with the kernels involving the derivatives of the scattered waves for point sources. Based on the mixed reciprocity principle, we propose two simple and feasible numerical schemes for reconstructing D-to-N map. Compared with the well-known obstacle boundary recovering schemes using the simulation of D-to-N map directly, the proposed schemes give the possible ways to realizing the probe methods using practical far-field data, with the advantage of no numerical differentiation for scattered wave in their implementations. We present some numerical examples for the D-to-N map, showing the validity and stability of our schemes.     

On the inverse problem for three-dimensional potential scattering

We consider a linear perturbation for the wave equation $\text{□u = 0}\text{in}\text{Ω = E}{}^3$ by “repulsive” smooth potentials q(y) that are small at infinity and suitably small (in magnitude). We use a time-dependent approach to prove that the scattering operator S(q) determines uniquenes uniquely the scatterer q (at least in this class). Energy inequalities will play a central role in our discussion.     

A solution method for the inverse scattering problem

We consider a numerical solution method for the inverse problem of quantum scattering theory based on Tikhonov's regularization method and Newton's method for solving a nonlinear operator equation of second kind.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 52–55, 1985.     

Shift-invariant spaces on the real line

We investigate the structure of shift-invariant spaces generated by a finite number of compactly supported functions in . Based on a study of linear independence of the shifts of the generators, we characterize such shift-invariant spaces in terms of the semi-convolutions of the generators with sequences on . Moreover, we show that such a shift-invariant space provides -approximation order if and only if it contains all polynomials of degree less than .

     

Fuzzy P-measure on the real line

The fuzzy measurable space on the real line, β_ϱ say, is defined by means of fuzzy relation ‘less or equal’ [5]. The main properties of β_ϱ are given. These results are used for investigation of fuzzy P-measure on β_ϱ. The principal emphasis is laid on the connections between a fuzzy P-measure and its cumulative distribution function. A theorem extending a fuzzy P-measure to β_ϱ is shown.     

Rational wavelets on the real line

In this note a representation of the discrete Green's function of a compact discretization of a two point boundary value problem of order n 2 is given which among other things allows a direct proff of the convergence (and divergence) properties.     

Isoperimetric problem on the real line

The isoperimetric problem on the real line for distributions with continuous positive densities is considered. Necessary and sufficient conditions under which the intervals (−∞, a) are extremal are suggested. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 5–9. Translated by V. Sudakov.     

Fourier interpolation on the real line

Publications mathématiques de l'IHÉS - In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a...     

Group topologies on the real line

It is shown that there exist 2^2c non-isomorphic group topologies on the real line, among them 2^2c are non-locally compact, ?₀ are compact (connected), ?₀ are locally compact and compactly generated. We also give some characterizations of the usual topology among group topologies on the real line.     

Colouring the real line

The problem of colouring the real line so that the distance between like coloured numbers does not lie in some specified set D, called the distance set, is discussed. In particular, the minimum number of colours needed for various distance sets are determined.