We study reverse triangle inequalities for Riesz potentials and their connection with polarization. This work generalizes inequalities for sup norms of products of polynomials, and reverse triangle inequalities for logarithmic potentials. The main to
ol used in the proofs is the representation for a power of the farthest distance function as a Riesz potential of a unit Borel measure.
LOFAR, the LOw-Frequency ARray, is a new-generation radio interferometer constructed in the north of the Netherlands and across europe. Utilizing a novel phased-array design, LOFAR covers the largely unexplored low-frequency range from 10-240 MHz and
provides a number of unique observing capabilities. Spreading out from a core located near the village of Exloo in the northeast of the Netherlands, a total of 40 LOFAR stations are nearing completion. A further five stations have been deployed throughout Germany, and one station has been built in each of France, Sweden, and the UK. Digital beam-forming techniques make the LOFAR system agile and allow for rapid repointing of the telescope as well as the potential for multiple simultaneous observations. With its dense core array and long interferometric baselines, LOFAR achieves unparalleled sensitivity and angular resolution in the low-frequency radio regime. The LOFAR facilities are jointly operated by the International LOFAR Telescope (ILT) foundation, as an observatory open to the global astronomical community. LOFAR is one of the first radio observatories to feature automated processing pipelines to deliver fully calibrated science products to its user community. LOFARs new capabilities, techniques and modus operandi make it an important pathfinder for the Square Kilometre Array (SKA). We give an overview of the LOFAR instrument, its major hardware and software components, and the core science objectives that have driven its design. In addition, we present a selection of new results from the commissioning phase of this new radio observatory.
Solitons are non-dispersing localized waves that occur in diverse physical settings. A variety of optical solitons have been observed, b
Utilizing $sim 50$ ks of Chandra X-ray Observatory imaging, we present an analysis of the intracluster medium (ICM) and cavity system in the galaxy cluster RBS 797. In addition to the two previously known cavities in the cluster core, the new and dee
per X-ray image has revealed additional structure associated with the active galactic nucleus (AGN). The surface brightness decrements of the two cavities are unusually large, and are consistent with elongated cavities lying close to our line-of-sight. We estimate a total AGN outburst energy and mean jet power of $approx 3 - 6 times 10^{60}$ erg and $approx 3 - 6 times 10^{45}$ erg s$^{-1}$, respectively, depending on the assumed geometrical configuration of the cavities. Thus, RBS 797 is apparently among the the most powerful AGN outbursts known in a cluster. The average mass accretion rate needed to power the AGN by accretion alone is $sim 1 M_{odot}$ yr$^{-1}$. We show that accretion of cold gas onto the AGN at this level is plausible, but that Bondi accretion of the hot atmosphere is probably not. The BCG harbors an unresolved, non-thermal nuclear X-ray source with a bolometric luminosity of $approx 2 times 10^{44}$ erg s$^{-1}$. The nuclear emission is probably associated with a rapidly-accreting, radiatively inefficient accretion flow. We present tentative evidence that star formation in the BCG is being triggered by the radio jets and suggest that the cavities may be driving weak shocks ($M sim 1.5$) into the ICM, similar to the process in the galaxy cluster MS 0735.6+7421.
We present a detailed study of soliton compression of ultra-short pulses based on phase-mismatched second-harmonic generation (textit{i.e.}, the cascaded quadratic nonlinearity) in bulk quadratic nonlinear media. The single-cycle propagation equation
s in the temporal domain including higher-order nonlinear terms are presented. The balance between the quadratic (SHG) and the cubic (Kerr) nonlinearity plays a crucial role: we define an effective soliton number -- related to the difference between the SHG and the Kerr soliton numbers -- and show that it has to be larger than unity for successful pulse compression to take place. This requires that the phase mismatch be below a critical level, which is high in a material where the quadratic nonlinearity dominates over the cubic Kerr nonlinearity. Through extensive numerical simulations we find dimensionless scaling laws, expressed through the effective soliton number, which control the behaviour of the compressed pulses. These laws hold in the stationary regime, in which group-velocity mismatch effects are small, and they are similar to the ones observed for fiber soliton compressors. The numerical simulations indicate that clean compressed pulses below two optical cycles can be achieved in a $beta$-barium borate crystal at appropriate wavelengths, even for picosecond input pulses.
