2D photonic crystal complete band gap search using a cyclic cellular automaton refination

We present a refination method based on a cyclic cellular automaton (CCA) that simulates a crystallization-like process, aided with a heuristic evolutionary method called differential evolution (DE) used to perform an ordered search of full photonic band gaps (FPBGs) in a 2D photonic crystal (PC). The solution is proposed as a combinatorial optimization of the elements in a binary array. These elements represent the existence or absence of a dielectric material surrounded by air, thus representing a general geometry whose search space is defined by the number of elements in such array. A block-iterative frequency-domain method was used to compute the FPBGs on a PC, when present. DE has proved to be useful in combinatorial problems and we also present an implementation feature that takes advantage of the periodic nature of PCs to enhance the convergence of this algorithm. Finally, we used this methodology to find a PC structure with a 19% bandgap-to-midgap ratio without requiring previous information of suboptimal configurations and we made a statistical study of how it is affected by disorder in the borders of the structure compared with a previous work that uses a genetic algorithm.     

Matrix decomposition algorithms for the finite element Galerkin method with piecewise Hermite cubics

Matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed for the solution of the systems of linear algebraic equations arising when the finite element Galerkin method with piecewise Hermite bicubics is used to solve Poisson’s equation on the unit square. Like their orthogonal spline collocation counterparts, these MDAs, which require O(N ²logN) operations on an N×N uniform partition, are based on knowledge of the solution of a generalized eigenvalue problem associated with the corresponding discretization of a two-point boundary value problem. The eigenvalues and eigenfunctions are determined for various choices of boundary conditions, and numerical results are presented to demonstrate the efficacy of the MDAs. Weiwei Sun was supported in part by a grant from City University of Hong Kong (Project No. CityU 7002110).     

Spectral theory of Volterra-composition operators

Given a Lebesgue measurable self-map of the interval [0, 1], the Volterra- composition operator is defined as We develop the spectral theory of these operators. In particular, for a class of natural symbols , finiteness of the spectrum is characterized and formulae for the trace and the convergence exponent of eigenvalues are provided. The positivity of the spectrum as well as the analyticity of the eigenfunctions are also treated. The theory of entire functions as well as solving some Cauchy Problems will play a fundamental role in this theory. We also supply some examples of symbols to which the theory can be applied and, in particular, eigenvalues and eigenfunctions are computed explicitly. Partially supported by Plan Nacional I+D+I grant no. MTM2006-09060, Junta de Andalucía FQM-260 and P06-FQM-02225.     

The Inverse Resonance Problem for Hermite Operators

In this paper the inverse resonance problem for the Hermite operator is investigated. The Hermite operator with the creation operator , the annihilation operator , and a finitely supported multiplication operator b, is an unbounded operator on ℓ ²(ℕ₀) having finitely many eigenvalues and infinitely many resonances (except for b=0, when there are no eigenvalues or resonances). It is shown that knowing the location of eigenvalues and resonances determines the potential b uniquely.      

Weyl and Lidski? Inequalities for General Hyperbolic Polynomials

The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices, after a conjecture by A. Horn. Among them are the so-called Weyl and Lidskiǐ inequalities. An elementary proof of the latter for hyperbolic polynomials is given. This proof follows an idea from H. Weinberger and is free from representation theory and Schubert calculus arguments, as well as from hyperbolic partial differential equations theory.     

Nontrivial solutions of Kirchhoff type problems

In this work, we study Kirchhoff type problems on a bounded domain. We consider the cases where the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity, and the nonlinearity is asymptotically linear near zero but 4-superlinear at infinity. By computing the relevant critical groups, we obtain nontrivial solutions via Morse theory.     

Lyapunov-type stability criterion for periodic generalized Camassa–Holm equations

We study the Lyapunov stability of the periodic generalized Camassa–Holm equation in terms of the periodic/anti-periodic eigenvalues and the associated spectral intervals. Moreover, we establish a Lyapunov-type stability criterion based on the Floquet theory and a Lyapunov-type inequality.     

Corrigendum to: “On the eigenvalues of operators with gaps. Application to Dirac operators” [J. Funct. Anal. 174 (1) (2000) 208–226]

In this corrigendum, we address some closability issues that were ignored in [1].