Article history: Received 26 November 2013 Received in revised form 20 February 2014 Accepted 31 March 2014 Available online 8 April 2014

Inequalities involving hyperbolic functions and trigonometric functions are established. Most of the established inequalities can be regarded as the companion inequalities to those obtained by Lazarević, Adamović and Mitrinović and Cusa and Huygens The established results are obtained with the aid of the Schwab-Borchardt mean.

We consider the Cauchy problem of the gravitational wave with the initial data distributed only around the brane in the one brane model of Randall and Sundrum, and examine its behavior as t → ∞ . Then we find its leading behavior is t unlike an ordinary flat 5-dimensional space-time. Such a signal shows that the Huygens principle is violated on the 4-dimensional brane world even asymptotically ...

This paper presents a study of the effects of symmetry on the generic bifurcation at a double-zero eigenvalue that was first investigated by Bogdanov and Takens. Two different symmetry groups are considered: Huygens symmetry and odd-Huygens symmetry. Here Huygens symmetry means that the system is equivariant under permutation of the two state variables. Using Hilbert-Weyl theory, normal forms a...

The scalar Huygens-Fresnel principle is reformulated to take into account the vector nature of light and its associated directed electric and magnetic fields. Based on Maxwell's equations, a vector Huygens secondary source is developed in terms of the fundamental radiating units of electromagnetism: the electric and magnetic dipoles. The formulation is in terms of the vector potential from whic...

A three-dimensional extension of the recently demonstrated generalization of the laws of refraction and reflection was investigated for both flat and curved metasurfaces. We found that out-of-plane refraction occurs for a metasurface that imparts a wavevector out of the plane of incidence onto the incident light beam. Metasurfaces provide arbitrary control over the direction of refraction, and ...

A terahertz vertical-external-cavity surface-emitting-laser (VECSEL) is demonstrated using an active focusing reflectarray metasurface based on quantum-cascade gain material. The focusing effect enables a hemispherical cavity with flat optics, which exhibits higher geometric stability than a plano-plano cavity and a directive and circular near-diffraction limited Gaussian beam with M2</sup...

We present a design of acoustic metasurfaces yielding asymmetric transmission within a certain frequency band. The design consists of a layer of gradient-index metasurface and a layer of low refractive index metasurface. Incident waves are controlled in a wave vector dependent manner to create strong asymmetric transmission. Numerical simulations show that the approach provides high transmissio...