We present an improved method for the precise reconstruction of cosmic-ray air showers above 10 17 eV with sparse radio arrays. The method is based on the comparison of measured pulses to predictions for radio pulse shapes by CoREAS simulations. We applied our method to the data of Tunka-Rex, a 1 km 2 radio array in Siberia operating in the frequency band of 30-80 MHz. Tunka-Rex is triggered by the air-Cherenkov detector Tunka-133 and by scintillators (Tunka-Grande). The instrument collects air-shower data since 2012. The present paper describes an updated data analysis of Tunka-Rex and details of the new method applied. After quality cuts, when Tunka-Rex reaches its full efficiency, the energy resolution of about 10% given by the new method has reached the limit of systematic uncertainties due to the calibration uncertainty and shower-to-shower fluctuations. At the same time the shower maximum reconstruction has improved compared to the previous method based on the slope of the lateral distribution and reaches a precision of better than 35 g/cm 2 . We also define conditions of the measurements at which the shower maximum resolution of Tunka-Rex reaches a value of 25 g/cm 2 and becomes competitive to optical detectors. To check and validate our reconstruction and efficiency cuts we compare individual events to the reconstruction of Tunka-133. Furthermore, we compare the mean of the shower maximum as a function of primary energy to the measurements of other experiments.PACS numbers: 96.50.sd, 95.55.Jz, 07.50.Qx,

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TAIGA stands for ``Tunka Advanced Instrument for cosmic ray physics and Gamma Astronomy'' and is a project to built a complex, hybrid detector system for ground-based gamma-ray astronomy from a few TeV to several PeV, and for cosmic ray studies from 100 TeV to 1 EeV. TAIGA will search for ``PeVatrons'' (ultra-high energy gamma-ray sources) and measure the composition and spectrum of cosmic rays in the knee region (100 TeV–10 PeV) with good energy resolution and high statistics. TAIGA will include Tunka-HiSCORE — an array of wide-angle air Cherenkov stations, an array of Imaging Atmospheric Cherenkov Telescopes, an array of particle detectors, both on the surface and underground and the TUNKA-133 air Cherenkov array.

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With the discovery of gamma ray bursts1,2, it became clear that our Universe flickers with superfast catastrophic events, sometimes lasting for a thousandths of a second. These ultra-fast transients - the peculiar one-day butterflies of the Universe - shine so brightly that they are noticed even on the other end of the Universe and, moreover, by very small telescopes. But in the radio range, the sky remained silent until the beginning of the 21st century. Only in 2007, radio astronomers analyzing archival observations of the Parkes Radio Telescope first encountered fast transients 3,4 . About a hundred such sources have already been discovered. We report the first optical observation of the closest radio burster FRB 180916.J0158+655-8 synchronously with a radio burst. In total, we obtained about 155,093 images at MASTER Global Robotic Net9*. In the course of our observations, we found a new method for detecting objects deep below the noise level. In addition, using the new method, we found the excess of photons in the FRB direction at a level of 23 m associated with the emission of the host galaxy.

A: The TAIGA observatory addresses ground-based gamma-ray astronomy at energies from a few TeV to several PeV, cosmic ray physics from 100 TeV to several EeV as well as for search for axion-like particles, Lorentz violations and another evidence of New Physics. In 2020 year a one square kilometer TAIGA setup should be put in operation.

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