Inverse Scattering

In this paper we present an improved algorithm to compute the minimal null-space basis of polynomial matrices, a problem which has many applications in control and systems theory. This algorithm takes advantage of the block Toeplitz structure of the Sylvester matrix associated with the polynomial matrix. The analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well-known pencil (state-space) algorithms found in the literature.

Ground Penetrating Radar (GPR) is one of the most feasible and friendly instrumentation to detect buried remains and perform diagnostics of archaeological structures with the aim of detecting hidden objects (defects, voids, constructive typology; etc..). In fact, GPR technique allows to perform measurements over large areas in a very fast way thanks to a portable instrumentation. Despite of the widespread exploitation of the GPR as data acquisition system, many difficulties arise in processing GPR data so to obtain images reliable and easily interpretable by the end-users. This difficulty is exacerbated when no a priori information is available as for example arises in the case of historical heritages for which the knowledge of the constructive modalities and materials of the structure might be completely missed. A possible answer to the above cited difficulties resides in the development and the exploitation of microwave tomography algorithms [1, 2], based on more refined electromagnetic scattering model with respect to the ones usually adopted in the classic radaristic approach. By exploitation of the microwave tomographic approach, it is possible to gain accurate and reliable "images" of the investigated structure in order to detect, localize and possibly determine the extent and the geometrical features of the embedded objects. In this framework, the adoption of simplified models of the electromagnetic scattering appears very convenient for practical and theoretical reasons. First, the linear inversion algorithms are numerically efficient thus allowing to investigate domains large in terms of the probing wavelength in a quasi real-time also in the case of 3D case also by adopting schemes based on the combination of 2D reconstruction . In addition, the solution approaches are very robust against the uncertainties in the parameters of the measurement configuration and on the investigated scenario. From a theoretical point of view, the linear models allow further advantages such as: the absence of the false solutions (a question to be arisen in non linear inverse problems); the exploitation of well known regularization tools for achieving a stable solution of the problem; the possibility to analyze the reconstruction performances of the algorithm once the measurement configuration and the properties of the host medium are known. Here, we will present the main features and the reconstruction results of a linear inversion algorithm based on the Born approximation in realistic applications in archaeology and cultural heritage diagnostics. Born model is useful when penetrable objects are under investigations. As well known, the Born Approximation is used to solve the forward problem, that is the determination of the scattered field from a known object under the hypothesis of weak scatterer, i.e. an object whose dielectric permittivity is slightly different from the one of the host medium and whose extent is small in term of probing wavelength. Differently, for the inverse scattering problem, the above hypotheses can be relaxed at the cost to renounce to a "quantitative reconstruction" of the object. In fact, as already shown by results in realistic conditions , the adoption of a Born model inversion scheme allows to detect, to localize and to determine the geometry of the object also in the case of not weak scattering objects. [1] R. Persico, R. Bernini, F. Soldovieri, "The role of the measurement configuration in inverse scattering from buried objects under the Born approximation", IEEE Trans.

Volume 1 The problem of moments, J. A. Shohat and J. D. Tamarkin 2 The theory of rings, N. Jacobson 3 Geometry of polynomials, M. Marden 4 The theory of valuations, O. F. G. Schilling 5 The kernel function and conformal mapping, S. Bergman 6 Introduction to the theory of algebraic functions of one variable, C. C. Chevalley 7.1 The algebraic theory of semigroups, Volume I, A. H. Clifford and G. B. Preston 7.2 The algebraic theory of semigroups, Volume II, A. H. Clifford and G. B. Preston 8 Discontinuous groups and automorphic functions, J. Lehner 9 Linear approximation, Arthur Sard 10 An introduction to the analytic theory of numbers, R. Ayoub 11 Fixed points and topological degree in nonlinear analysis, J. Cronin 12 Uniform spaces, J. R. Isbell

Using the inverse-scattering transform with 3X3 U-V matrix representation and fully exploiting the symmetry properties of the scattering matrix elements, we found the one-parameter single-soliton, the four-parameter breather soliton, and the general ¹oliton solutions of a perturbed nonlinear Schrodinger equation which describes the femtosecond pulse propagation in optical 6bers. The thresh- old power below which the one-parameter single soliton cannot be formed was given. The main characteristic of the general single-soliton solution of the perturbed nonlinear Schrodinger equation is that it presents an arbitrary number of "humps" (loca1 maxima of the amplitude) of different heights.

The quality of seismic images obtained by reverse time migration ͑RTM͒ strongly depends on the imaging condition. We propose a new imaging condition that is motivated by stationary phase analysis of the classical crosscorrelation imaging condition. Its implementation requires the Poynting vector of the source and receiver wavefields at the imaging point. An obliquity correction is added to compensate for the reflector dip effect on amplitudes of RTM. Numerical experiments show that using an imaging condition with obliquity compensation improves reverse time migration by reducing backscattering artifacts and improving illumination compensation.

Let q(r) , r = |x| , x ∈ R 3 , be a real-valued square-integrable compactly supported function, and [0, a] be the smallest interval containing the support of q(r) . Let A(α , α) = A(α • α) be the corresponding scattering amplitude at a fixed positive energy, k 2 = 1 . Let δ be the phase = a , provided that q(r) does not change sign in some, arbitrary small, neighborhood of a .

Let q(r), r=|x|, , be a real-valued square-integrable compactly supported function, and be the smallest interval containing the support of q(r). Let be the corresponding scattering amplitude at a fixed positive energy, . Let be the phase shifts at k = 1. It is proved that , provided that q(r) does not change sign in some, arbitrary small, neighbourhood of a.

The ab initio No-Core Shell Model (NCSM) begins with an intrinsic Hamiltonian for all nucleons in the nucleus. Realistic two-nucleon and tri-nucleon interactions are incorporated such as those recently developed from effective field theory and chiral perturbation theory. We then derive a finite basis-space dependent Hermitian effective Hamiltonian that conserves all the symmetries of the initial Hamiltonian. The resulting finite Hamiltonian matrix problem is solved by diagonalization on parallel computers. Applications range from light nuclei to multiquark systems and, recently, to quantum field theory including systems with bosons. We present this approach with a sample of recent results.

A procedure for constructing bound state potentials is given. We show that, under the natural conditions imposed on a radial eigenvalue problem, the only special cases of the general central potential, which are exactly solvable and have infinite number of energy eigenvalues, are the Coulomb and harmonic oscillator potentials.

In this paper, we consider a non linear epidemic model SIQS by experiencing the disease; whenever infected, the disease individuals will return to the susceptible class after a fixed period of time. First, the local stabilities and global stability of the infection-free equilibrium and endemic equilibrium are analyzed, respectively. Second, the endemic equilibrium is formulated in terms of the incidence rate, and after we stydy local and global asymptotic stability. We study the stochastique systme by pertubing the contact rate, and we mainly use the theorie of Itô formula. Finnaly we study the spatial spread model with traveling wave solution.

We investigate the relation between the local variables of a discrete integrable lattice system and the corresponding separation variables, derived from the associated spectral curve. In particular, we have shown how the inverse transformation from the separation variables to the discrete lattice variables may be factorised as a sequence of canonical transformations, following the procedure outlined by Kuznetsov.

Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surfaces and obtain their bifurcations in generic 1-parameter families of such surfaces.

The Davydov model for exciton–phonon coupling in hydrogen-bonded molecular chains is reconsidered in the context of two-wave resonant interaction. By applying a semi-discrete slowly varying envelope approximation, when the physical problem is that of the long-distance evolution of an input finite duration excitonic pulse, we derive an integrable limit model which preserves the coupling nature of the process. The spectral transform is constructed with emphasis on the complete characterization of the spectral data. As an application, the localized one-soliton solution is explicitly constructed. Then by using Darboux–Bäcklund transformations, a non-local (or topological) one-soliton solution is also derived. As a consequence, the system possess two different soliton solutions where the phonon component is a localized pulse, but where the exciton wave is either localized (bell shape) or topological (kink shape). The resulting approximate soliton solutions of the Davydov model in the re...

With the aid of a modified similarity transformation we have obtained exact energy eigenvalues of the generalized Dirac -Coulomb equation. This equation consists of the time component of the Lorentz 4-vector potential V v (r) = -A 1 /r, and a Lorentz scalar potential V s (r) = -A 2 /r. The transformed radial equations are so simple so that their solutions are inferred from the conventional solutions of the Schrödinger -Coulomb equation.