Deformations of Annuli on Riemann surfaces with Smallest Mean Distortion

Let $A$ and $A'$ be two circular annuli and let $\rho$ be a radial metric defined in the annulus $A'$. Consider the class $\mathcal H_\rho$ of $\rho-$harmonic mappings between $A$ and $A'$. It is proved recently by Iwaniec, Kovalev and Onninen that, if $\rho=1$ (i.e. if $\rho$ is Euclidean metric) then $\mathcal H_\rho$ is not empty if and only if there holds the Nitsche condition (and thus is proved the J. C. C. Nitsche conjecture). In this paper we formulate an condition (which we call $\rho-$Nitsche conjecture) with corresponds to $\mathcal H_\rho$ and define $\rho-$Nitsche harmonic maps. We determine the extremal mappings with smallest mean distortion for mappings of annuli w.r. to the metric $\rho$. As a corollary, we find that $\rho-$Nitsche harmonic maps are Dirichlet minimizers among all homeomorphisms $h:A\to A'$. However, outside the $\rho$-Nitsche condition of the modulus of the annuli, within the class of homeomorphisms, no such energy minimizers exist. % However, %outside the $\rho-$Nitsche range of the modulus of the annuli, %within the class of homeomorphisms, no such energy minimizers exist. This extends some recent results of Astala, Iwaniec and Martin (ARMA, 2010) where it is considered the case $\rho=1$ and $\rho=1/|z|$.Comment: Some misprints are corrected in this version (see Lemma~5.1

Common Codebook Millimeter Wave Beam Design: Designing Beams for Both Sounding and Communication with Uniform Planar Arrays

Fifth generation (5G) wireless networks are expected to utilize wide bandwidths available at millimeter wave (mmWave) frequencies for enhancing system throughput. However, the unfavorable channel conditions of mmWave links, e.g., higher path loss and attenuation due to atmospheric gases or water vapor, hinder reliable communications. To compensate for these severe losses, it is essential to have a multitude of antennas to generate sharp and strong beams for directional transmission. In this paper, we consider mmWave systems using uniform planar array (UPA) antennas, which effectively place more antennas on a two-dimensional grid. A hybrid beamforming setup is also considered to generate beams by combining a multitude of antennas using only a few radio frequency chains. We focus on designing a set of transmit beamformers generating beams adapted to the directional characteristics of mmWave links assuming a UPA and hybrid beamforming. We first define ideal beam patterns for UPA structures. Each beamformer is constructed to minimize the mean squared error from the corresponding ideal beam pattern. Simulation results verify that the proposed codebooks enhance beamforming reliability and data rate in mmWave systems.Comment: 14 pages, 10 figure

Strongly modulated transmission of a spin-split quantum wire with local Rashba interaction

We investigate the transport properties of ballistic quantum wires in the presence of Zeeman spin splittings and a spatially inhomogeneous Rashba interaction. The Zeeman interaction is extended along the wire and produces gaps in the energy spectrum which allow electron propagation only for spinors lying along a certain direction. For spins in the opposite direction the waves are evanescent far away from the Rashba region, which plays the role of the scattering center. The most interesting case occurs when the magnetic field is perpendicular to the Rashba field. Then, the spins of the asymptotic wavefunctions are not eigenfunctions of the Rashba Hamiltonian and the resulting coupling between spins in the Rashba region gives rise to sudden changes of the transmission probability when the Fermi energy is swept along the gap. After briefly examining the energy spectrum and eigenfunctions of a wire with extended Rashba coupling, we analyze the transmission through a region of localized Rashba interaction, in which a double interface separates a region of constant Rashba interaction from wire leads free from spin-orbit coupling. For energies slightly above the propagation threshold, we find the ubiquitous occurrence of transmission zeros (antiresonances) which are analyzed by matching methods in the one-dimensional limit. We find that a a minimal tight-binding model yields analytical transmission lineshapes of Fano antiresonance type. More general angular dependences of the external magnetic field is treated within projected Schroedinger equations with Hamiltonian matrix elements mixing wavefunction components. Finally, we consider a realistic quantum wire where the energy subbands are coupled via the Rashba intersubband coupling term and discuss its effect on the transmission zeros.Comment: 11 pages, 12 figure