Some formulae for the q-Bernoulli and Euler polynomials of higher order

The purpose of this paper is to give a proof of Kummer type congruence for the q-Bernoulli numbers of higher order, which is an answer to a part of the problem in a previous publication (see Indian J. Pure Appl. Math. 32 (2001) 1565-1570).     

Three-dimensional radiative transfer in an anisotropically scattering, semi-infinite medium: generalized reflection function

Three-dimensional radiative transfer in an anisotropic scattering medium exposed to spatially varying, collimated radiation is studied. The generalized reflection function for a semi-infinite medium with a very general scattering phase function is the focus of this investigation. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to formulate a nonlinear integral equation for the generalized reflection function. The integration is over both the polar and azimuthal angles; hence, the integral equation is said to be in the double-integral form. The double-integral, reflection function formulation can handle a variety of anisotropic phase functions and does not require an expansion of the phase function in a Legendre polynomial series. Complicated kernel transformations of previous single-integral studies are eliminated. Single and double scattering approximations are developed. Numerical results are presented for a Rayleigh phase function to illustrate the computational characteristics of the method and are compared to results obtained with the single-integral method. Agreement between the two approaches is excellent; however, as the transform variable increases beyond five the number of quadrature points required for the double-integral method to produce accurate solutions significantly increases. A new interpolation scheme produces accurate results when the transform variable is large.     

A comprehensive study of the long pulse Nd:YAG laser drilling of multi-layer carbon fibre composites

The results of an extensive experimental study of the free running Nd:YAG laser drilling of a multi-layer carbon fibre composite, where adjacent layers have differently orientated fibres, are reported. For holes drilled with the laser operating in fixed-Q mode at 1064 nm, parallel sections of blind holes illustrating discontinuities in the hole size along a given section direction will be shown to occur at the interface between adjacent layers. An explanation for this effect is proposed. Detailed single pulse drilling characteristics will be presented illustrating the exit hole diameter as a function of pulse energy and material thickness. These characteristics illustrate a ‘stable' drilling regime in which the exit hole diameters are least sensitive to changes in pulse energy or material thickness and a less ‘stable' regime in which they are more strongly dependent on these parameters. Drilling characteristics will be given for two different beam qualities, illustrating the greater drilling depth and reduced hole size achievable with an improved beam quality. Finally holes drilled through a 2 mm thick sample of material with multiple pulses are considered. Size distribution curves for entrance and exit holes will be presented. The total energy required (number of pulses × pulse energy) to drill through 2 mm thick material will be reported as a function of pulse energy in stationary air and argon atmospheres and in a partial vacuum, illustrating a threshold energy which is dependent upon the drilling atmosphere. The threshold energies will be discussed with reference to plasma formation and the reactivity of the drilling atmosphere.     

Rb saturation spectroscopy with an optically narrowed FM diode laser

A fast-frequency modulated (FM) diode laser has been optically narrowed using the technique of resonant optical feedback, to provide linewidths in each FM mode of ≈ 200 kHz peak-to-peak. With a drive frequency of 50 MHz and modulation index of 0.2, the FM laser has been used for the first time to obtain Doppler-free FM spectra of the Rb D₁ line at 795 nm. The potential use of this system for laser frequency stabilisation is discussed.     

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Truncated Newton method for sparse unconstrained optimization using automatic differentiation

When solving large complex optimization problems, the user is faced with three major problems. These are (i) the cost in human time in obtaining accurate expressions for the derivatives involved; (ii) the need to store second derivative information; and (iii), of lessening importance, the time taken to solve the problem on the computer. For many problems, a significant part of the latter can be attributed to solving Newton-like equations. In the algorithm described, the equations are solved using a conjugate direction method that only needs the Hessian at the current point when it is multiplied by a trial vector. In this paper, we present a method that finds this product using automatic differentiation while only requiring vector storage. The method takes advantage of any sparsity in the Hessian matrix and computes exact derivatives. It avoids the complexity of symbolic differentiation, the inaccuracy of numerical differentiation, the labor of finding analytic derivatives, and the need for matrix store. When far from a minimum, an accurate solution to the Newton equations is not justified, so an approximate solution is obtained by using a version of Dembo and Steihaug's truncated Newton algorithm (Ref. 1).This paper was presented at the SIAM National Meeting, Boston, Massachusetts, 1986.     

3D-TOFSIMS characterization of black spots in polymer light emitting diodes

The occurrence and formation of black spots areas in PolyLED devices has been studied by time-of-flight SIMS (TOFSIMS). The composition, shape and position of the black spots is visualised by three-dimensional (3D)-TOFSIMS depth-profiling. It has been established that the formation of non-emissive spots is due to the growth of aluminium oxide clusters at the AlBa/polymer interface. Electron injection in the black spots is lost by the resulting local increase of the resistivity of the cathode.     

Rehabilitation of the Gauss-Jordan algorithm

Summary In this paper a Gauss-Jordan algorithm with column interchanges is presented and analysed. We show that, in contrast with Gaussian elimination, the Gauss-Jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability. For solutions obtained by Gauss-Jordan with column interchanges, a more satisfactory bound for the residual norm can be given. The analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by Gaussian elimination and that, in most practical situations, the residuals are equally small. This is confirmed by numerical experiments. Moreover, timing experiments on a Cyber 205 vector computer show that the algorithm presented has good vectorisation properties.