Steepest Descent, CG, and Iterative Regularization of Ill-Posed Problems

The state of the art iterative method for solving large linear systems is the conjugate gradient (CG) algorithm. Theoretical convergence analysis suggests that CG converges more rapidly than steepest descent. This paper argues that steepest descent may be an attractive alternative to CG when solving linear systems arising from the discretization of ill-posed problems. Specifically, it is shown that, for ill-posed problems, steepest descent has a more stable convergence behavior than CG, which may be explained by the fact that the filter factors for steepest descent behave much less erratically than those for CG. Moreover, it is shown that, with proper preconditioning, the convergence rate of steepest descent is competitive with that of CG.This revised version was published online in October 2005 with corrections to the Cover Date.     

Comparative studies of femtosecond to microsecond laser pulses on selective pigmented cell injury in skin

Threshold radiant exposures for grossly apparent immediate whitening and ultrastructural alterations of melanosomes in black guinea pig skin were determined for a series of red visible laser pulses ranging from 4 x 10(-4) to 6.5 x 10(-14) s. Threshold exposures for melanosomal injury were found to be independent of pulsewidth when the pulsewidths were below the estimated thermal relaxation time of melanosomes. Threshold radiant exposures for melanosomal injury were found to increase when the pulsewidths were approximately equal to or above the thermal relaxation time of melanosomes. At longer pulse durations, fracturing of melanosomes was not observed despite the longer exposures necessary for injury. Instead, perimelanosomal vacuoles were noted. These findings are consistent with the theory of selective photothermolysis and provide evidence for the thermal initiation of melanosomal disruption.     

Two-dimensional autocorrelation function analysis of smoked foil patterns

Digital image processing techniques have been applied to the analysis of cellular smoked foil patterns from gaseous detonations. In particular, the two-dimensional autocorrelation function is applied to digital cell pattern images and an orientational correlation parameter is calculated. Taking line profiles along the directions of highest correlation provides an unbiased method of determining the mean cell size in each of the two principal directions. By analyzing the width, amplitude and angular position of the orientational correlation plots, information can be extracted concerning the cellular pattern regularity, the relative angular correlation between two sets of transverse waves in two directions, and the mean shape or elongation of the cells within the pattern. The technique is applied to smoked foils from oxyacetylene mixtures with argon dilutions ranging from 0 to 75% to quantify the increase in regularity with argon dilution. This method provides a simple and useful way of analyzing cellular patterns and constitutes a promising technique for improving smoked foil diagnostics.     

Asymptotic properties of computationally efficient alternative estimators for a class of multivariate normal models

Parameters of Gaussian multivariate models are often estimated using the maximum likelihood approach. In spite of its merits, this methodology is not practical when the sample size is very large, as, for example, in the case of massive georeferenced data sets. In this paper, we study the asymptotic properties of the estimators that minimize three alternatives to the likelihood function, designed to increase the computational efficiency. This is achieved by applying the information sandwich technique to expansions of the pseudo-likelihood functions as quadratic forms of independent normal random variables. Theoretical calculations are given for a first-order autoregressive time series and then extended to a two-dimensional autoregressive process on a lattice. We compare the efficiency of the three estimators to that of the maximum likelihood estimator as well as among themselves, using numerical calculations of the theoretical results and simulations.     

Ihara zeta functions of digraphs

After defining and exploring some of the properties of Ihara zeta functions of digraphs, we improve upon Kotani and Sunada’s bounds on the poles of Ihara zeta functions of undirected graphs by considering digraphs whose adjacency matrices are directed edge matrices.     

Lyapunov-type inequalities for nonlinear systems

In this paper, by using elementary analysis, we establish some new Lyapunov-type inequalities for nonlinear systems of differential equations, special cases of which contain the well-known equations such as Emden-Fowler-type and half-linear equations. The inequalities obtained here can be used as handy tools in the study of qualitative behaviour of solutions of the associated equations.     

Romanovski polynomials in selected physics problems

We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark physics to random matrix theory. It appears timely to draw attention to it by the present study. Our survey also includes several new observations on the orthogonality properties of the Romanovski polynomials and new developments from their Rodrigues formula.