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We present an efficient null space free Jacobi–Davidson method to compute the positive eigenvalues of time harmonic Maxwell’s equations. We focus on a class of spatial discretizations that guarantee the existence of discrete vector potentials, such as Yee’s scheme and the edge elements. During the Jacobi–Davidson iteration, the correction process is applied to the vector potential instead. The correction equation is solved approximately as in the standard Jacobi–Davidson approach. The computational cost of the transformation from the vector potential to the corrector is negligible. As a consequence, the expanding subspace automatically stays out of the null space and no extra projection step is needed. Numerical evidence confirms that the proposed scheme indeed outperforms the standard and projection-based Jacobi–Davidson methods by a significant margin.

Knotted ribbons form an important topic in knot theory. They have applications in natural sciences, such as cyclic duplex DNA modeling. A flat knotted ribbon can be obtained by gently pulling a knotted ribbon tight so that it becomes flat and folded. An important problem in knot theory is to study the minimal ratio of length to width of a flat knotted ribbon. This minimal ratio is called the ribbonlength of the knot. It has been conjectured that the ribbonlength has an upper bound and a lower bound which are both linear in the crossing number of the knot. In the first part of the paper, we use grid diagrams to construct flat knotted ribbons and prove an explicit quadratic upper bound on the ribbonlength for all non-trivial knots. We then improve the quadratic upper bound to a linear upper bound for all non-trivial torus knots and twist knots. Our approach of using grid diagrams to study flat knotted ribbons is novel and can likely be used to obtain a linear upper bound for more general families of knots. In the second part of the paper, we obtain a sharper linear upper bound on the ribbonlength for nontrivial twist knots by constructing a flat knotted ribbon via folding the ribbon over itself multiple times to shorten the length.

No abstract uploaded!