In this paper we propose adaptive finite element methods for computing the band structure of 2D periodic photonic crystals and of photonic crystal fibres, modelled as spectral problems for Maxwell’s equations under either TM or TE polarisation. With the application of the Floquet transform the problem of computing the spectrum can be reduced to the computation of the discrete spectra of each me...

Let B be a real vector lattice and a Banach space under a semimonotonic norm. Suppose T is a linear operator on B which is positive and eventually compact, y is a positive vector, and A is a positive real. It is shown that (XI—TY1y is positive if, and only if, y is annihilated by the absolute value of any generalized eigenvector of T* associated with a strictly positive eigenvalue not less than...

We study a robust and efficient eigensolver for computing the positive dense spectrum of the two-dimensional transmission eigenvalue problem (TEP) which is derived from the Maxwell’s equation with complex media in pseudo-chiral model and the transverse magnetic mode. The discretized governing equations by the Nédélec edge element result in a large-scale quadratic eigenvalue problem (QEP). We es...

The multigroup neutron SP N equations, which are an approximation of the transport equation, used to model nuclear reactor cores. In their steady state, these equations can be written as a source problem or eigenvalue problem. We study resolution those two problems with H 1 -conforming finite element method and Discontinuous Galerkin method, namely Symmetric Interior Penalty method.

We consider an eigenvalue problem for a certain type of quasi-linear second-order di®erential equation on the interval ð0;1Þ. Using an appropriate version of the mountain pass theorem, we establish the existence of a positive solution in H 1 0 ð0;1Þ for a range of values of the eigenvalue. It is shown that these solutions generate solutions of Maxwell's equations having the form of guided trave...

Hopf bifurcations in problems with O(2) symmetry are considered. In these problems, the Jacobian matrix is always singular at the circle of Z2 symmetric steady state solutions. While a couple of imaginary eigenvalue cross the imaginary axis, the Hopf bifurcation is not of standard type. The canonical coordinates transformation is used for removing the zero eigenvalue and converting the problem ...

Equilibria in mechanics or in transportation models are not always expressed through a system of equations, but sometimes they are characterized by means of complementarity conditions involving a convex cone. This work deals with the analysis of cone-constrained eigenvalue problems. We discuss some theoretical issues like, for instance, the estimation of the maximal number of eigenvalues in a c...

We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of...

The Seiberg-Witten equations are studied from the viewpoint of gauge potential decomposition. We find a determinant equation ∆Aμ = −λAμ for the twisting U(1) potential Aμ of the Seiberg-Witten theory, which is in itself an eigenvalue problem of the Laplacian operator, with the eigenvalue being the vacuum expectation value of the field function, λ = ‖Φ‖ /2. This establishes a direct relationship...

We present efficient numerical techniques for calculation of eigenvalue distributions of random matrices in the beta-ensembles. We compute histograms using direct simulations on very large matrices, by using tridiagonal matrices with appropriate simplifications. The distributions are also obtained by numerical solution of the Painlevé II and V equations with high accuracy. For the spacings we s...