The purpose of the paper is twofold. First, we show that partial-data transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in $\mathbb{R}^n$, $n=2,3$. Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [26] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [26] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [26] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.

The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems.

Precise characterization of concealed person-worn objects will speed up the passenger screening process by reducing the rate of nuisance alarms, while also enhancing the airport security imaging systems. This paper presents an automatic, real-time method for wideband millimeter-wave radar identification of the nominal surface contours of the human body – even with affixed foreign objects or when a segment of the body cross-section is not captured by the radar – without relying on the body's bilateral symmetry. The developed method is verified experimentally when applied to the actual images generated by a laboratory airport scanning prototype developed recently by the US Department of Homeland Security (DHS). Our method uses the noisy collection of radar cross-section reflectivity data to extract the main contours and estimates the nominal body surface cross-sections through fitting a small-term Fourier series of circumferential variation. This is a necessary step for accurate characterizing of concealed terrorist threat objects affixed to the body.

This paper presents a new artificial neural network (ANN)-based radar data processing approach for the detection of breast cancers located inside fibro-glandular tissues. The aim is not the breast imaging but detecting tumors through ANN processing of data extracted from the radar signals measured around the breast. The proposed approach has been assessed using several realistic two-dimensional breast geometries derived from the models provided by the numerical breast phantom repository of the University of Wisconsin Cross-Disciplinary Electromagnetic Laboratory (UWCEM). The pulsed radar system was assumed to operate in the mono-static configuration. The obtained results showed the abilities of the proposed approach to detect, for any single radar trace, tumors located inside the fibro-glandular tissues with a sensitivity of 93%, a specificity of 90%, and an overall accuracy of 92%.

We propose a reliable direct imaging method based on the reverse time migration for finding extended obstacles with phaseless total field data. We prove that the imaging resolution of the method is essentially the same as the imaging results using the scattering data with full phase information when the measurement is far away from the obstacle. The imaginary part of the cross-correlation imaging functional always peaks on the boundary of the obstacle. Numerical experiments are included to illustrate the powerful imaging quality

Consider the time-harmonic acoustic scattering from an extended elastic body surrounded by a finite number of point-like obstacles in a fluid. We assume point source waves are emitted from arrayed transducers and the signals of scattered near-field data are recorded by receivers not far away from the scatterers (compared to the incident wavelength). The forward scattering can be modeled as an interaction problem between acoustic and elastic waves together with a multiple scattering problem between the extend solid and point scatterers. We prove a necessary and sufficient condition that can be used simultaneously to recover the shape of the extended elastic solid and to locate the positions of point scatterers. The essential ingredient in our analysis is the outgoing-to-incoming (OtI) operator applied to the resulting near-field response matrix (or operator). In the first part, we justify the MUSIC algorithm for locating point scatterers from near-field measurements. In the second part, we apply the factorization method, the continuous analogue of MUSIC, to the two-scale scattering problem for determining both extended and point scatterers. Numerical examples in 2D are demonstrated to show the validity and accuracy of our inversion algorithms.

A novel method is developed for solving the inverse problem of reconstructing the shape of an interior cavity. The boundary of the cavity is assumed to be a small and smooth perturbation of a circle. The incident field is generated by a point source inside the cavity. The scattering data is taken on a circle centered at the source. The method requires only a single incident wave at one frequency. Using a transformed field expansion, the original boundary value problem is reduced to a successive sequence of two-point boundary value problems and is solved in a closed form. By dropping higher order terms in the power series expansion, the inverse problem is linearized and an explicit relation is established between the Fourier coefficients of the cavity surface function and the total field. A nonlinear correction algorithm is devised to improve the accuracy of the reconstruction. Numerical results are presented to show the effectiveness of the method and its ability to obtain subwavelength resolution.

Consider the Schrödinger operator −∇2 + q with a smooth compactly supportedpotential q,q =q(x),x ∈R3.

Let A(β,α,k)be the corresponding scattering amplitude, k2 be the energy, α ∈S2 be the incident direction,β ∈S2 be the direction of scattered wave,S2 be the unit sphere in R3. Assume thatk =k0> 0 is fixed, andα =α0 is fixed. Then the scattering data areA(β) =A(β,α0,k0)= Aq(β)is a function on S2. The following inverse scatteringproblem is studied: IP: Given an arbitrary f ∈L2(S2)and an arbitrary small number ϵ> 0, can one find q ∈ C0∞(D) , where D ∈R3 is an arbitrary fixed domain, suchthat ||Aq(β) −f(β)||L2(S2)<ϵ? A positive answer to this question is given. A method for constructing such aq isproposed. There are infinitely many such q, not necessarily real-valued.

Inverse scattering problem is discussed for the Maxwell’s equations. A reduction of theMaxwell’s system to a new Fredholm second-kind integral equation with a scalarweakly singular kernel is given for electromagnetic (EM) wave scattering. Thisequation allows one to derive a formula for the scattering amplitude in which only ascalar function is present. If this function is small (an assumption that validates aBorn-type approximation), then formulas for the solution to the inverse problem areobtained from the scattering data: the complex permittivityϵ′(x) in a bounded regionD ⊂ R3 is found from the scattering amplitudeA(β,α,k) known for a fixed k = ω √ϵ0μ0 >0 and allβ,α ∈ S2, whereS2 is the unit sphere in R3,ϵ0 and μ0 are constantpermittivity and magnetic permeability in the exterior regionD′ = R3\D. The novel pointsin this paper include: i) A reduction of the inverse problem for vectorEM waves to a vector integral equation with scalar kernelwithout any symmetry assumptions on the scatterer, ii) A derivation of thescalar integral equation of the first kind for solving the inversescattering problem, and iii) Presenting formulas for solving this scalar integralequation. The problem of solving this integral equation is an ill-posed one. A method fora stable solution of this problem is given.

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic. The main goal of this paper is to introduce a Riemann–Hilbert problem whose solution gives the solution of our initial–boundary-value problem. This is a preliminary step to obtain uniform long-time asymptotics for the solution of this equation using both the stationary phase and the Deift–Zhou methods.

We show that the support of a (possibly) coated anisotropic medium is uniquely determined by the electric far-field patterns corresponding to incident time-harmonic electromagnetic plane waves with arbitrary polarization and direction. Our proof avoids the use of a fundamental solution to Maxwell’s equations in an anisotropic medium and instead relies on the well-posedness and regularity properties of solutions to an interior transmission problem for Maxwell’s equations.

AMS 2000 Mathematics subject classification: Primary 35R30; 35Q60. Secondary 35P25; 78A45