Dyadic Sets, Maximal Functions and Applications on ax+bax+b --Groups

Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure $\rho$, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,\rho)$ admits a Calder\'on--Zygmund decomposition which involves a particular family of sets, called Calder\'on--Zygmund sets. In this paper, we first show the existence of a dyadic grid in the group $S$, which has {nice} properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid we shall prove a Fefferman--Stein type inequality, involving the dyadic maximal Hardy--Littlewood function and the dyadic sharp dyadic function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space $H^1$ and the $BMO$ space introduced in [Collect. Math. 60(2009), 277--295].Comment: Math. Z. (to appear

Frequency reconfigurable patch antenna for 4G LTE applications

A compact printed multi-band frequency reconfigurable patch antenna for 4G LTE applications is presented in this paper (50 x 60 x 1.6 mm3). The antenna consists of W-shaped and Inverted-U shaped patch lines connected in a Tree-shape on the front side of the antenna. The back-side of the antenna contains a 90°-tilted T-shaped strip connected with an Inverted-L shaped strip which is shorted with a patch on the front side for increasing the electrical length to cover lower frequency bands. Frequency reconfigurability is achieved by inserting three switches i.e., PIN diodes. The most critical part of this work is the designing of RLC-based DC line circuits for providing the DC biasing to the PIN diodes used as switches and inserting them at optimum locations. This antenna is reconfigurable among eight different 4G LTE frequency bands including 0.9 GHz, 1.4 GHz, 1.5 GHz, 1.6 GHz, 1.7 GHz, 1.8 GHz, 2.6 GHz, 3.5 GHz and WLAN band 2.5 GHz. The antenna exhibits different radiation patterns having a different direction of peak gain at different frequencies and for different switching combinations. The antenna is simulated with CST, and a prototype is fabricated to compare the measured and simulated results with good accuracy

Photospheric Electric Fields and Energy Fluxes in the Eruptive Active Region NOAA 11158

How much electromagnetic energy crosses the photosphere in evolving solar active regions? With the advent of high-cadence vector magnetic field observations, addressing this fundamental question has become tractable. In this paper, we apply the "PTD-Doppler-FLCT-Ideal" (PDFI) electric field inversion technique of Kazachenko et al. (2014) to a 6-day HMI/SDO vector magnetogram and Doppler velocity sequence, to find the electric field and Poynting flux evolution in active region NOAA 11158, which produced an X2.2 flare early on 2011 February 15. We find photospheric electric fields ranging up to $2$ V/cm. The Poynting fluxes range from $[-0.6$ to $2.3]\times10^{10}$ ergs$\cdot$cm$^{-2}$s$^{-1}$, mostly positive, with the largest contribution to the energy budget in the range of $[10^9$-$10^{10}]$ ergs$\cdot$cm$^{-2}$s$^{-1}$. Integrating the instantaneous energy flux over space and time, we find that the total magnetic energy accumulated above the photosphere from the initial emergence to the moment before the X2.2 flare to be $E=10.6\times10^{32}$ ergs, which is partitioned as $2.0$ and $8.6\times10^{32}$ ergs, respectively, between free and potential energies. Those estimates are consistent with estimates from preflare non-linear force-free field (NLFFF) extrapolations and the Minimum Current Corona estimates (MCC), in spite of our very different approach. This study of photospheric electric fields demonstrates the potential of the PDFI approach for estimating Poynting fluxes and opens the door to more quantitative studies of the solar photosphere and more realistic data-driven simulations of coronal magnetic field evolution.Comment: 51 pages, 10 figures, accepted by ApJ on August 11, 201

Equilibrium states for sectional-hyperbolic attractors

It has been conjectured that the original Lorenz attractor supports a unique measure of maximal entropy. In this article, we give a partial answer to this question (and its higher-dimensional counterpart) by considering the uniqueness of equilibrium states for H\"older continuous functions on a sectional-hyperbolic attractor $\Lambda$. We prove that $C^1$-open and densely, if the point masses at singularities are not equilibrium states, then there exists a unique equilibrium state supported on $\Lambda$. In particular, there exists a unique measure of maximal entropy for the flow $X|_\Lambda$