An efficient semi-analytic method is developed for computing the band structures of two-dimensional photonic crystals which are triangular lattices of circular cylinders. The problem is formulated as an eigenvalue problem for a given frequency using the Dirichlet-to-Neumann (DtN) map of a hexagon unit cell. This is a linear eigenvalue problem even if the material is dispersive, where the eigenv...

The problem of nding the smallest eigenvalue and the corresponding eigenspace of a symmetric matrix is stated as a semide nite optimization problem. A straightforward application of nowadays more or less standard routines for the solution of semide nite problems yields a new algorithm for the smallest eigenvalue problem; the approach not only yields the smallest eigenvalue, but also a symmetric...

We present a new computational approach for a class of large-scale nonlinear eigenvalue problems (NEPs) that are nonlinear in the eigenvalue. The contribution of this paper is two-fold. We derive a new iterative algorithm for NEPs, the tensor infinite Arnoldi method (TIAR), which is applicable to a general class of NEPs, and we show how to specialize the algorithm to a specific NEP: the wavegui...

A new method is proposed for designing IIR digital allpass filters with an equiripple phase response that can be proven to be optimal in the Chebyshev sense. The proposed procedure is based on the formulation of an eigenvalue problem by using the Remez exchange algorithm. Since there exists more than one eigenvalue in the general eigenvalue problem, we introduce a new and very simple selection ...

The low-rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations. To exploit the low-rank damping property, we propose a Padé Approximate Linearization (PAL) algorithm. The advantage of the PAL algorithm is that the dimension of the resulting linear eigenvalue problem is only n+ lm, which is generally substantially smaller than the dimension 2n of th...

In this lecture we abstract the eigenvalue problems that we have found so useful thus far for solving the PDEs to a general class of boundary value problems that share a common set of properties. The so-called Sturm-Liouville Problems define a class of eigenvalue problems, which include many of the previous problems as special cases. The S − L Problem helps to identify those assumptions that ar...

We describe a necessary condition for zero-eigenvalue Turing instability, i.e., Turing instability arising from a real eigenvalue changing sign from negative to positive, for general chemical reaction networks modeled with mass-action kinetics. The reaction mechanisms are represented by the species-reaction graph (SR graph), which is a bipartite graph with different nodes representing species a...

This paper presents a sequential semidefinite programming (SDP) approach to maximize the minimal eigenvalue of the generalized eigenvalue problem, in which the two symmetric matrices defining the eigenvalue problem are supposed to be the polynomials in terms of the variables. An important application of this problem is found in the structural optimization which attempts to maximize the minimal ...

It was conjectured by Doron and Smilansky (DS) in 1992 that k2 > 0 is a Dirichlet eigenvalue of the Laplacian in a bounded domain D if and only if the corresponding S-matrix for the scattering problem by the obstacle D has an eigenvalue 1. The main results of this paper are: 1) a proof that the DS conjecture is incorrect, 2) a proof that a necessary condition for 1 to be an eigenvalue of the S-...

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...