On the Hecke Eigenvalues of Maass Forms

Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\alpha _{p},\;\beta _{p}$ are the Satake parameters of $\phi$ at $p$, and $c$ is an absolute constant with $0<c<1$. In fact, $c$ can be taken as $0.27332$. In addition, we prove that the natural density of such primes $p$ ($p\nmid N$ and $|\alpha_{p}|=|\beta_{p}| = 1$) is at least $34/35$.Comment: Version 2: typos corrected and a new section on natural density adde

Spectral characterization of a specific class of trees

In this paper, it is shown that the graph $T_4(p,q,r)$ is determined by its Laplacian spectrum and there are no two non-isomorphic such graphs which are cospectral with respect to adjacency spectrum.Comment: 11 pages, 4 figure

Superlensing properties of one-dimensional dielectric photonic crystals

We present the experimental observation of the superlensing effect in a slab of a one-dimensional photonic crystal made of tilted dielectric elements. We show that this flat lens can achieve subwavelength resolution in different frequency bands. We also demonstrate that the introduction of a proper corrugation on the lens surface can dramatically improve both the transmission and the resolution of the imaged signal.Comment: 9 pages, 9 figure