On the density of certain spectral points for a class of $ C^{2} $ quasiperiodic Schrödinger cocycles

<p style='text-indent:20px;'>For <inline-formula><tex-math id="M2">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> cos-type potentials, large coupling constants, and fixed <inline-formula><tex-math id="M3">\begin{document}$ Diophantine $\end{document}</tex-math></inline-formula> frequency, we show that the density of the spectral points associated with the Schrödinger operator is larger than 0. In other words, for every fixed spectral point <inline-formula><tex-math id="M4">\begin{document}$ E $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \liminf\limits_{\epsilon\to 0}\frac{|(E-\epsilon,E+\epsilon)\bigcap\Sigma_{\alpha,\lambda\upsilon}|}{2\epsilon} = \beta $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \beta\in [\frac{1}{2},1] $\end{document}</tex-math></inline-formula>. Our approach is a further improvement on the papers [<xref ref-type="bibr" rid="b15">15</xref>] and [<xref ref-type="bibr" rid="b17">17</xref>].</p>

Effect of longitudinal epsilon-near-zero regime in dynamics of ultrashort laser pulses in hyperbolic metamaterials

Optical properties of hyperbolic metamaterials (HMMs) are in stark contrast to properties of ordinary media that fuels interest to various applications of HMMs in photonics. Special attention is attributed to the epsilon-near zero regime (ENZ) of HMMs that is the spectral point in which real part of the permittivity of the HMM becomes zero. This is accompanied by the effects of field enhancement having far-reaching applications. Here we focus on the experimental and theoretical investigation of the propagation of an ultrashort laser pulse through the silver nanorod-based HMM slab in the spectral range over the ENZ. We revealed pronounced resonant change of the pulse delay in HMMs and the transition between the superluminal and slow pulse propagation at the ENZ spectral point. Observed dynamical phenomena are confirmed theoretically and attributed to unusual case when the spectral half of an ultrashort pulse has elliptical dispersion and another has the hyperbolic one. Special attention is payed to the propagation of chirped laser pulses in the HMMs.

A New k-Distribution Scheme for Clear-Sky Radiative Transfer Calculations in Earth’s Atmosphere. Part I: Thermal Infrared (Longwave) Radiation

A new k-distribution scheme of longwave radiation without the correlated-k-distribution assumption is developed. Grouping of spectral points is based on the line-by-line (LBL)-calculated absorption coefficient k at a few sets of reference pressure pr and temperature θr, where the cooling rate is substantial in a spectral band. In this new scheme, the range of k(pr, θr) of a band is divided into a number of equal intervals, or g groups, in log10(kr). A spectral point at the wavenumber ν is identified with one of the g groups according to its kν(pr, θr). For each g group, a Planck-weighted k-distribution function Hg and a nonlinearly averaged absorption coefficient k¯g⁡(p,θ) are derived. The function Hg and the absorption coefficient k¯g⁡(p,θ) constitute the new k-distribution scheme. In this k-distribution scheme, a spectral point can only be identified with a g group regardless of pressure and temperature, which is different from the correlated-k distribution scheme. The k-distribution scheme is applied to the H2O, CO2, O3, N2O, and CH4 absorption bands, and results are compared with LBL calculations. To balance between the accuracy and the computational economy, the number of g groups in a band of a given gas is chosen such that 1) the difference in cooling rate is &lt;0.1 K day−1 in the troposphere and &lt;1.0 K day−1 in the stratosphere and 2) the difference in fluxes is &lt;0.5 W m−2 at both the top of the atmosphere and the surface. These differences are attained with 130 g groups, which is the sum of the g groups of all five gases.