Stability Properties of the Time Domain Electric Field Integral Equation Using a Separable Approximation for the Convolution with the Retarded Potential

The state of art of time domain integral equation (TDIE) solvers has grown by leaps and bounds over the past decade. During this time, advances have been made in (i) the development of accelerators that can be retrofitted with these solvers and (ii) understanding the stability properties of the electric field integral equation. As is well known, time domain electric field integral equation solvers have been notoriously difficult to stabilize. Research into methods for understanding and prescribing remedies have been on the uptick. The most recent of these efforts are (i) Lubich quadrature and (ii) exact integration. In this paper, we re-examine the solution to this equation using (i) the undifferentiated form of the TD-EFIE and (ii) a separable approximation to the spatio-temporal convolution. The proposed scheme can be constructed such that the spatial integrand over the source and observer domains is smooth and integrable. As several numerical results will demonstrate, the proposed scheme yields stable results for long simulation times and a variety of targets, both of which have proven extremely challenging in the past.Comment: 9 pages, 13 figures. To be published in IEEE Transactions on Antennas and Propagatio

Electric field distortions in structures of the twist bend nematic (NTB) phase of a bent-core liquid crystal

Dielectric spectroscopy of a twist bend nematic phase of an achiral bent core liquid crystalline compound under DC bias is used to investigate its response to electric field. Two collective relaxation processes are revealed, these are assigned to distortions of helicoidal structure by the external bias field. Frequency of the mode depends primarily on the helicoidal angle and has anomalous, softening- like behaviour at the nematic to the twist bend nematic transition. A coupling of dielectric anisotropy with electric field gives rise to a new equilibrium periodic structure in the time scale involved. The modulus of the wave vector gradually vanishes on increasing the bias field (except for the initial behaviour, which is just the opposite). Transition from the twist bend to the splay bend structure is clearly observed by a sudden drop in the frequency of this mode, which decreases almost linearly with increasing field. Results agree with predictions from current models for the periodically distorted a twist bend nematic phase.Comment: 14 PAGES, 7 FIGURES, submitted to Physical Review Letter

Scale invariant correlations and the distribution of prime numbers

Negative correlations in the distribution of prime numbers are found to display a scale invariance. This occurs in conjunction with a nonstationary behavior. We compare the prime number series to a type of fractional Brownian motion which incorporates both the scale invariance and the nonstationary behavior. Interesting discrepancies remain. The scale invariance also appears to imply the Riemann hypothesis and we study the use of the former as a test of the latter.Comment: 13 pages, 8 figures, version to appear in J. Phys.