Lower bound of overall exponential sums for polynomialsϕ(x d )

LetS ⁽¹⁾ (n, Q) denote the maximum module of exponential sums for polynomials of degree LE" BORDER="0">n over the Galois fieldF _Q . In a previous paper the transition to the multiple exponential sums allowed us to obtain a good lower bound of the valueS ⁽¹⁾ (n, Q), which coincides with Weil's bound whenn = q ^(m-1)/2 + 1, whereq, m are odd andm LE" BORDER="0"> 3. Here the same approach is used for the estimation of the valueS ^(d) (n, Q), which corresponds to polynomialsLE" BORDER="0">(x ^d ) overF _Q , whered is any divisor ofq – 1.     

Uniform fourth order difference scheme for a singular perturbation problem

Summary The numerical solution of a nonlinear singularly perturbed two-point boundary value problem is studied. The developed method is based on Hermitian approximation of the second derivative on special discretization mesh. Numerical examples which demonstrate the effectiveness of the method are presented.This research was partly supported by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.-Yugoslav Joint Board on Scientific and Technological Cooperation (grants JF 544, JF 799)     

Lattice-point examples for a question of Erdős

Here we establish a set of eight points in general position in the plane, i.e. no three on a line, no four on a circle, and they determine 7 distinct distances, so that, thei-th distance occursi times,i = 1, 2, LE" BORDER="0">, 7. The points are embedded in a triangular net, and the distances are not ordered by size or in any other way. We shall show that some known and unknown examples forn < 8 with the above properties may also be lattice points of a similar net.Research (partially) supported by the Hungarian National Foundation for Scientific Research (OTKA) grant, no. 1808.     

A fourth order method for a singular two-point boundary value problem

Recently, Chawla et al. described a second order finite difference method for the class of singular two-point boundary value problems:

Stability properties of collocation methods

Lower bounds for LINE" BORDER="0"> are given for which equidistant s-point collocation methods areA(LINE" BORDER="0">)-stable for arbitrarys.     

Adaptive boundary element methods for strongly elliptic integral equations

Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.This work was carried out while Dr. De-hao Yu was an LE" BORDER="0">Alexander-von-Humboldt-StiftungLE" BORDER="0"> research fellow at the University of Stuttgart in 1987, 1988     

What do multistep methods approximate?

Summary Standard analysis of multistep methods for ODE's assumes the application of an initialization routine that generates the starting points. Here ak-step method is considered directly as a mappingR ^kn LINE" BORDER="0">R ⁿ . It is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field. Some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution.     

Convolution quadrature and discretized operational calculus. I

Numerical methods are derived for problems in integral equations (Volterra, Wiener-Hopf equations) and numerical integration (singular integrands, multiple time-scale convolution). The basic tool of this theory is the numerical approximation of convolution integrals

On the generation and annealing of dangling bond defects in hydrogenated amorphous silicon

We show that, through the diffusive re-arrangement of Si-H bonds, the a-SiLINE" BORDER="0">H lattice is able to establish thermal equilibrium between the densities of band tail trapped charge carriers and dangling bond defects. When this equilibrium is disturbed by changes in temperature, carrier injection or illumination, dangling bond defects have to be generated or annealed out via H-diffusion processes. Based on the concept of charge-induced bond breaking, we develop a mathematical formalism for the diffusive re-arrangement of Si-H bonds and show that our formalism can account for a variety of observations that have been made in the context of defect-generation and annealing experiments.     

Influence of end reflectivity on gain saturation in a planar distributed feedback laser

An approximate method for the analysis of planar-waveguide distributed-feedback lasers is extended to include a nonvanishing reflectivity at the ends of the structure. An index grating structure with parasitic losses is investigated. An expression for the small-signal gain coefficient LINE" BORDER="0">^0MN as a function of the output power P ^MNout normalized to the saturation power P ^s , coupling coefficient K, complex reflectivity R of end reflectors, and waveguide parameters is presented.