On the Second-order Frechet Derivatives of Eigenvalues of Sturm-Liouville Problems in Potentials

The works of V. A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm-Liouville problems are analytic in potentials, considered as mappings from the Lebesgue space to the space of real numbers and the Banach space of continuous functions respectively. Moreover, the first-order Frechet derivatives are known and paly an important role in many problems. In this paper, we will find the second-order Frechet derivatives of eigenvalues in potentials, which are also proved to be negative definite quadratic forms for some cases.Comment: 10 page

Symmetry selective third harmonic generation from plasmonic metacrystals

Nonlinear processes are often governed by selection rules imposed by the symmetries of the molecular configurations. The most well-known examples include the role of mirror symmetry breaking for the generation of even harmonics, and the selection rule related to the rotation symmetry in harmonic generation for fundamental beams with circular polarizations. While the role of mirror symmetry breaking in second harmonic generation has been extensively studied in plasmonic systems, the investigation on selection rules pertaining to circular polarization states of harmonic generation has been limited to crystals, i.e. symmetries at the atomic level. Here we demonstrate the rotational symmetry dependent third harmonic generation from nonlinear plasmonic metacrystals. We show that the selection rule can be imposed by the rotational symmetry of meta-crystals embedded into an isotropic organic nonlinear thin film. The results presented here may open new avenues for designing symmetry-dependent nonlinear optical responses with tailored plasmonic nanostructures.Comment: 13 pages, 3 figure

Comparison of Sampling Designs for Estimating Deforestation from Landsat TM and MODIS Imagery: A Case Study in Mato Grosso, Brazil

Sampling designs are commonly used to estimate deforestation over large areas, but comparisons between different sampling strategies are required. Using PRODES deforestation data as a reference, deforestation in the state of Mato Grosso in Brazil from 2005 to 2006 is evaluated using Landsat imagery and a nearly synchronous MODIS dataset. The MODIS-derived deforestation is used to assist in sampling and extrapolation. Three sampling designs are compared according to the estimated deforestation of the entire study area based on simple extrapolation and linear regression models. The results show that stratified sampling for strata construction and sample allocation using the MODIS-derived deforestation hotspots provided more precise estimations than simple random and systematic sampling. Moreover, the relationship between the MODIS-derived and TM-derived deforestation provides a precise estimate of the total deforestation area as well as the distribution of deforestation in each block