Distribution function estimates by Wasserstein metric and Bernstein approximation for C-1 functions

The aim of the paper is to estimate the density functions or distribution functions measured by Wasserstein metric, a typical kind of statistical distances, which is usually required in the statistical learningBased on the classical Bernstein approximation, a scheme is presented.To get the error estimates of the scheme, the problem turns to estimating the L1 norm of the Bernstein approximation for monotone C-1functions, which was rarely discussed in the classical approximation theoryFinally, we get a probability estimate by the statistical distance.     

Korn inequality on irregular domains

In this paper, we study the weighted Korn inequality on some irregular domains, e.g., s-John domains and domains satisfying quasihyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality.     

Enhancement of single‑photon emission from nitrogen‑vacancy centers with TiN/(Al,Sc)N hyperbolic metamaterial

The broadband enhancement of single‑photon emission from nitrogen‐vacancy centers in nanodiamonds coupled to a planar multilayer metamaterial with hyperbolic dispersion is studied experimentally. The metamaterial is fabricated as an epitaxial metal/dielectric superlattice consisting of CMOS‐compatible ceramics: titanium nitride (TiN) and aluminum scandium nitride (Al_xSc_1‐xN). It is demonstrated that employing the metamaterial results in significant enhancement of collected single‑photon emission and reduction of the excited‐state lifetime. Our results could have an impact on future CMOS‐compatible integrated quantum sources.

     

The Two-Dimensional Electrides XONa (X=Mg,Ca) as Novel Natural Hyperbolic Materials

In two-dimensional electrides, anionic electrons are spatially confined in interlayer regions with high density, comparable to metals, and they are highly mobile, just as free electrons, resembling hyperbolic metamaterials with metal-dielectric multilayered structures. In this work, two-dimensional electride materials MgONa and CaONa are proposed as good natural hyperbolic materials. By using the first-principles calculations based on density functional theory (DFT), the electronic structures, stabilities, and optical properties of two-dimensional electride materials XONa (X=Mg, Ca) are investigated. Our results show that they are stable in 1-monolayer (1-ML) structures as well as in bulk states. They exhibit hyperbolic dispersions from visible to near infrared spectral range with high qualities up to about 700, which is two orders-of-magnitude larger than the preceding bulk hyperbolic materials. Numerical results reveal that they exhibit negative refraction with low losses. Their high-quality hyperbolic responses over a wide spectral range pave the way of broad photonic applications as natural hyperbolic materials.     

Strichartz Estimates for Non-Elliptic Schrödinger Equations on Compact Manifolds

In this note we consider the Schrödinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-Gérard-Tzvetkov to the non-elliptic case and improves the result by Salort for the degenerate case. We also investigate the optimality of the result for the case on 𝕊³ × 𝕊³.     

Theoretic quantum information entropies for the generalized hyperbolic potential

The Shannon entropy (S) and the Fisher Information (I) entropies are investigated for a generalized hyperbolic potential in position and momentum spaces. First, the Schrodinger equation is solved exactly using the Nikiforov-Uvarov-Functional Analysis method to obtain the energy spectra and the corresponding wave function. By Fourier transforming the position space wave function, the corresponding momentum wave function was obtained for the low-lying states corresponding to the ground and first excited states. The positions and momentum of Shannon entropy and Fisher Information entropies were calculated numerically. Finally, the Bialynicki-Birula and Mycielski and the Stam-Cramer-Rao inequalities for the Shannon entropy and Fisher Information entropies, respectively, were tested and were found to be satisfied for all cases considered.     

非等熵气体动力学方程组大初值问题的放缩框架

非等熵气体动力学系统Cauchy问题弱解全局存在性有两个公开问题:一个是包含真空的小初值问题,另一个是任意大初值问题.本文通过引入一个放缩框架证明了上述两个问题的等价性,即对于粘性消失解,其包含真空小初值问题的一致BV估计蕴含着任意大初值问题弱解的全局存在性.该放缩框架对大多数具有物理背景的双曲守恒律系统亦成立.