Electromagnetic vortex lines riding atop null solutions of the Maxwell equations

New method of introducing vortex lines of the electromagnetic field is outlined. The vortex lines arise when a complex Riemann-Silberstein vector $({\bm E} + i{\bm B})/\sqrt{2}$ is multiplied by a complex scalar function $\phi$. Such a multiplication may lead to new solutions of the Maxwell equations only when the electromagnetic field is null, i.e. when both relativistic invariants vanish. In general, zeroes of the $\phi$ function give rise to electromagnetic vortices. The description of these vortices benefits from the ideas of Penrose, Robinson and Trautman developed in general relativity.Comment: NATO Workshop on Singular Optics 2003 To appear in Journal of Optics

Shear-Free Gravitational Waves in an Anisotropic Universe

We study gravitational waves propagating through an anisotropic Bianchi I dust-filled universe (containing the Einstein-de-Sitter universe as a special case). The waves are modeled as small perturbations of this background cosmological model and we choose a family of null hypersurfaces in this space-time to act as the histories of the wavefronts of the radiation. We find that the perturbations we generate can describe pure gravitational radiation if and only if the null hypersurfaces are shear-free. We calculate the gauge-invariant small perturbations explicitly in this case. How these differ from the corresponding perturbations when the background space-time is isotropic is clearly exhibited.Comment: 32 pages, accepted for publication in Physical Review

A second derivative SQP method: theoretical issues

Sequential quadratic programming (SQP) methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems. Although second derivative information may often be calculated, there is little practical theory that justifies exact-Hessian SQP methods. In particular, the resulting quadratic programming (QP) subproblems are often nonconvex, and thus finding their global solutions may be computationally nonviable. This paper presents a second-derivative SQP method based on quadratic subproblems that are either convex, and thus may be solved efficiently, or need not be solved globally. Additionally, an explicit descent-constraint is imposed on certain QP subproblems, which “guides” the iterates through areas in which nonconvexity is a concern. Global convergence of the resulting algorithm is established

A second derivative SQP method: local convergence

In [19], we gave global convergence results for a second-derivative SQP method for minimizing the exact ℓ1-merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm. \ud \ud Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positive-definite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positive-definite matrix Bk—a simple diagonal approximation and a more sophisticated limited-memory BFGS update. We also analyze a strategy for updating the penalty paramter based on approximately minimizing the ℓ1-penalty function over a sequence of increasing values of the penalty parameter.\ud \ud Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the so-called Maratos effect. We show that a nonmonotone varient of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set

The re-emission spectrum of digital hardware subjected to EMI

The emission spectrum of digital hardware under the influence of external electromagnetic interference is shown to contain information about the interaction of the incident energy with the digital circuits in the system. The generation mechanism of the re-emission spectrum is reviewed, describing how nonlinear effects may be a precursor to the failure of the equipment under test. Measurements on a simple circuit are used to demonstrate how the characteristics of the re-emission spectrum may be correlated with changes to the digital waveform within the circuit. The technique is also applied to a piece of complex digital hardware where Similar, though more subtle, effects can be measured. It is shown that the re-emission spectrum can be used to detect the interaction of the interference with the digital devices at a level well below that which is able to cause static failures in the circuits. The utility of the technique as a diagnostic tool for immunity testing of digital hardware, by identifying which subsystems are being affected by external interference, is also demonstrated