Scattering Polarization in the Presence of Magnetic and Electric Fields

The polarization of radiation by scattering on an atom embedded in combined external quadrupole electric and uniform magnetic fields is studied theoretically. Limiting cases of scattering under Zeeman effect and Hanle effect in weak magnetic fields are discussed. The theory is general enough to handle scattering in intermediate magnetic fields (Hanle-Zeeman effect) and for arbitrary orientation of magnetic field. The quadrupolar electric field produces asymmetric line shifts and causes interesting level-crossing phenomena either in the absence of an ambient magnetic field or in its presence. It is shown that the quadrupolar electric field produces an additional depolarization in the $Q/I$ profiles and rotation of the plane of polarization in the $U/I$ profile over and above that arising from magnetic field itself. This characteristic may have a diagnostic potential to detect steady state and time varying electric fields that surround radiating atoms in Solar atmospheric layers.Comment: 41 pages, 6 figure

Combinatorial integer labeling theorems on finite sets with applications

Tucker’s well-known combinatorial lemma states that, for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {±1, ±2, · · · , ±n} with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper, we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set {±1, ±2, · · · , ±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state that, under some mild conditions, there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set {0, 1} n +q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided

Parallel algorithms for matrix polynomial division

AbstractIn this paper, several algorithms for the matrix polynomial division are taken into consideration. Such algorithms represent extensions of known parallel algorithms for the scalar polynomial division with remainder. The interest resides in the comparison of the parallel computational cost of these algorithms in the general non scalar case

Local Error Estimates in Quadrature

A theoretical error estimate for quadrature formulas, which depends on four approximations of the integral, is derived. We obtain a bound, often sharper than the trivial one, which requires milder conditions to be satisfied than a similar result previously presented by Laurie. A selection of numerical tests with one-dimensional integrals is reported, to show how the error estimate works in practice

Preconditioners based on fit techniques for the iterative regularization in the image deconvolution problem

For large-scale image reconstruction problems, the iterative regularization methods can be favorable alternatives to the direct methods. We analyze preconditioners for regularizing gradient-type iterations applied to problems with 2D band Toeplitz coefficient matrix. For problems having separable and positive definite matrices, the fit preconditioner we have introduced in a previous paper has been shown to be effective in conjunction with CG. The cost of this preconditioner is of O(n^2) operations per iteration, where n^2 is the pixels number of the image, whereas the cost of the circulant preconditioners commonly used for this type of problems is of O(n^2 log n) operations per iteration. In this paper the extension of the fit preconditioner to more general cases is proposed: namely the nonseparable positive definite case and the symmetric indefinite case. The major difficulty encountered in this extension concerns the factorization phase, where a further approximation is required. Three approximate factorizations are proposed. The preconditioners thus obtained have still a cost of O(n^2) operations per iteration. A numerical experimentation shows that the fit preconditioners Are competitive with the regularizing Chan preconditioner, both in the regularing efficiency and the computational cost

Regularizing Inverse Preconditioners for Symmetric Band Toeplitz Matrices

<p/> <p>Image restoration is a widely studied discrete ill-posed problem. Among the many regularization methods used for treating the problem, iterative methods have been shown to be effective. In this paper, we consider the case of a blurring function defined by space invariant and band-limited PSF, modeled by a linear system that has a band block Toeplitz structure with band Toeplitz blocks. In order to reduce the number of iterations required to obtain acceptable reconstructions, in 13 an inverse Toeplitz preconditioner for problems with a Toeplitz structure was proposed. The cost per iteration is of <inline-formula><graphic file="1687-6180-2007-085606-i1.gif"/></inline-formula> operations, where <inline-formula><graphic file="1687-6180-2007-085606-i2.gif"/></inline-formula> is the pixel number of the 2D image. In this paper, we propose inverse preconditioners with a band Toeplitz structure, which lower the cost to <inline-formula><graphic file="1687-6180-2007-085606-i3.gif"/></inline-formula> and in experiments showed the same speed of convergence and reconstruction efficiency as the inverse Toeplitz preconditioner.</p