We present a superalgebraically convergent integral equation algorithm for evaluation of TE and TM electromagnetic scattering by smooth perfectly conducting periodic surfaces z = f͑x͒. For grating-diffraction problems in the resonance regime (heights and periods up to a few wavelengths) the proposed algorithm produces solutions with full double-precision accuracy in single-processor computing times of the order of a few seconds. The algorithm can also produce, in reasonable computing times, highly accurate solutions for very challenging problems, such as (a) a problem of diffraction by a grating for which the peak-to-trough distance equals 40 times its period that, in turn, equals 20 times the wavelength; and (b) a high-frequency problem with very small incidence, up to 0.01°from glancing. The algorithm is based on the concurrent use of Floquet and Chebyshev expansions together with certain integration weights that are computed accurately by means of an asymptotic expansion as the number of integration points tends to infinity.

The compression of blood vessels by surrounding tissue is an important problem in hemodynamics, most prominently in studies relating to the heart. In this study we consider a long tube of elliptic cross section as an idealization of the geometry of a compressed blood vessel. An exact solution of the governing equations for pulsatile flow in a tube of elliptic cross section involves Mathieu functions which are considerably more difficult to evaluate than the Bessel functions in the case of a circular cross section. Results for the velocity field, flow rate and wall shear stress are obtained for different values of the pulsation frequency and ellipticity, with emphasis on how the effects of frequency and ellipticity combine to determine the flow characteristics. It is found that in general the effects of ellipticity are minor when frequency is low but become highly significant as the frequency increases. More specifically, the velocity profile along the major axis of the elliptic cross section develops sharp double peaks; the flow rate is reduced in approximately the same proportion as in the case of circular cross section; and the point of maximum shear on the tube wall migrates away from the minor axis where it is located in steady flow.

Abstract. We consider the problem of evaluating the current distribution J(z) that is induced on a straight wire antenna by a time-harmonic incident electromagnetic field. The scope of this paper is twofold. One of its main contributions is a regularity proof for a straight wire occupying the interval [−1, 1]. In particular, for a smooth time-harmonic incident field this theorem implies thatis an infinitely differentiable function-the previous state of the art in this regard placed I in the Sobolev space W 1,p , p > 1. The second focus of this work is on numerics: we present three superalgebraically convergent algorithms for the solution of wire problems, two based on Hallén's integral equation and one based on the Pocklington integrodifferential equation. Both our proof and our algorithms are based on two main elements: (1) a new decomposition of the kernel of the form G(z) = F 1 (z) ln|z| + F 2 (z), where F 1 (z) and F 2 (z) are analytic functions on the real line; and (2) removal of the end-point square root singularities by means of a coordinate transformation. The Hallén-and Pocklington-based algorithms we propose converge superalgebraically: faster than O(N −m ) and O(M −m ) for any positive integer m, where N and M are the numbers of unknowns and the number of integration points required for construction of the discretized operator, respectively. In previous studies, at most the leading-order contribution to the logarithmic singular term was extracted from the kernel and treated analytically, the higher-order singular derivatives were left untreated, and the resulting integration methods for the kernel exhibit O(M −3 ) convergence at best. A rather comprehensive set of tests we consider shows that, in many cases, to achieve a given accuracy, the numbers N of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available; the required number M of integration points, in turn, can be several orders of magnitude smaller than those required in previous methods. In particular, four-digit solutions were found in computational times of the order of four seconds and, in most cases, of the order of a fraction of a second on a contemporary personal computer; much higher accuracies result in very small additional computing times.

It is shown that Einstein's field equations for all perfect-fluid k = 0 FLRW cosmologies have the same form as the topological normal form of a fold bifurcation. In particular, we assume that the cosmological constant is a bifurcation parameter, and as such, fold bifurcation behaviour is shown to occur in a neighbourhood of Minkowski spacetime in the phase space. We show that as this cosmological constant parameter is varied, an expanding and contracting de Sitter universe emerge via this bifurcation.