783 research outputs found
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
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.
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