Topological transversals to a family of convex sets

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say that $\mathcal F$ has a \emph{topological $\rho$-transversal of index $(m,k)$} ($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal $m$-planes to $\mathcal F$ as $m$-planes containing a fixed $\rho$-plane in $\mathbb R^{m+k}$. Clearly, if $\mathcal F$ has a $\rho$-transversal plane, then $\mathcal F$ has a topological $\rho$-transversal of index $(m,k),$ for $\rho<m$ and $k\leq d-m$. The converse is not true in general. We prove that for a family $\mathcal F$ of $\rho+k+1$ compact convex sets in $\mathbb R^d$ a topological $\rho$-transversal of index $(m,k)$ implies an ordinary $\rho$-transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences

Ultraviolet and Infrared Divergences in Implicit Regularization: a Consistent Approach

Implicit Regularization is a 4-dimensional regularization initially conceived to treat ultraviolet divergences. It has been successfully tested in several instances in the literature, more specifically in those where Dimensional Regularization does not apply. In the present contribution we extend the method to handle infrared divergences as well. We show that the essential steps which rendered Implicit Regularization adequate in the case of ultraviolet divergences have their counterpart for infrared ones. Moreover we show that a new scale appears, typically an infrared scale which is completely independent of the ultraviolet one. Examples are given.Comment: 9 pages, version to appear in Mod. Phys. Lett.

Tunka-Rex: the Cost-Effective Radio Extension of the Tunka Air-Shower Observatory

Tunka-Rex is the radio extension of the Tunka cosmic-ray observatory in Siberia close to Lake Baikal. Since October 2012 Tunka-Rex measures the radio signal of air-showers in coincidence with the non-imaging air-Cherenkov array Tunka-133. Furthermore, this year additional antennas will go into operation triggered by the new scintillator array Tunka-Grande measuring the secondary electrons and muons of air showers. Tunka-Rex is a demonstrator for how economic an antenna array can be without losing significant performance: we have decided for simple and robust SALLA antennas, and we share the existing DAQ running in slave mode with the PMT detectors and the scintillators, respectively. This means that Tunka-Rex is triggered externally, and does not need its own infrastructure and DAQ for hybrid measurements. By this, the performance and the added value of the supplementary radio measurements can be studied, in particular, the precision for the reconstructed energy and the shower maximum in the energy range of approximately $10^{17}-10^{18}\,$eV. Here we show first results on the energy reconstruction indicating that radio measurements can compete with air-Cherenkov measurements in precision. Moreover, we discuss future plans for Tunka-Rex.Comment: Proceeding of UHECR 2014, Springdale, Utah, USA, accepted by JPS Conference Proceeding

Radio measurements of the energy and the depth of the shower maximum of cosmic-ray air showers by Tunka-Rex

We reconstructed the energy and the position of the shower maximum of air showers with energies $E \gtrsim 100$PeV applying a method using radio measurements performed with Tunka-Rex. An event-to-event comparison to air-Cherenkov measurements of the same air showers with the Tunka-133 photomultiplier array confirms that the radio reconstruction works reliably. The Tunka-Rex reconstruction methods and absolute scales have been tuned on CoREAS simulations and yield energy and $X_{\mathrm{max}}$ values consistent with the Tunka-133 measurements. The results of two independent measurement seasons agree within statistical uncertainties, which gives additional confidence in the radio reconstruction. The energy precision of Tunka-Rex is comparable to the Tunka-133 precision of $15$, and exhibits a $20$ uncertainty on the absolute scale dominated by the amplitude calibration of the antennas. For $X_{\mathrm{max}}$, this is the first direct experimental correlation of radio measurements with a different, established method. At the moment, the $X_{\mathrm{max}}$ resolution of Tunka-Rex is approximately $40$g/cm$^2$. This resolution can probably be improved by deploying additional antennas and by further development of the reconstruction methods, since the present analysis does not yet reveal any principle limitations.Comment: accepted for publication by JCA

Fourier-Space Crystallography as Group Cohomology

We reformulate Fourier-space crystallography in the language of cohomology of groups. Once the problem is understood as a classification of linear functions on the lattice, restricted by a particular group relation, and identified by gauge transformation, the cohomological description becomes natural. We review Fourier-space crystallography and group cohomology, quote the fact that cohomology is dual to homology, and exhibit several results, previously established for special cases or by intricate calculation, that fall immediately out of the formalism. In particular, we prove that {\it two phase functions are gauge equivalent if and only if they agree on all their gauge-invariant integral linear combinations} and show how to find all these linear combinations systematically.Comment: plain tex, 14 pages (replaced 5/8/01 to include archive preprint number for reference 22

Signal recognition and background suppression by matched filters and neural networks for Tunka-Rex

The Tunka Radio Extension (Tunka-Rex) is a digital antenna array, which measures the radio emission of the cosmic-ray air-showers in the frequency band of 30-80 MHz. Tunka-Rex is co-located with TAIGA experiment in Siberia and consists of 63 antennas, 57 of them are in a densely instrumented area of about 1 km\textsuperscript{2}. In the present work we discuss the improvements of the signal reconstruction applied for the Tunka-Rex. At the first stage we implemented matched filtering using averaged signals as template. The simulation study has shown that matched filtering allows one to decrease the threshold of signal detection and increase its purity. However, the maximum performance of matched filtering is achievable only in case of white noise, while in reality the noise is not fully random due to different reasons. To recognize hidden features of the noise and treat them, we decided to use convolutional neural network with autoencoder architecture. Taking the recorded trace as an input, the autoencoder returns denoised trace, i.e. removes all signal-unrelated amplitudes. We present the comparison between standard method of signal reconstruction, matched filtering and autoencoder, and discuss the prospects of application of neural networks for lowering the threshold of digital antenna arrays for cosmic-ray detection.Comment: ARENA2018 proceeding

Current Status and New Challenges of The Tunka Radio Extension

The Tunka Radio Extension (Tunka-Rex) is an antenna array spread over an area of about 1~km$^2$. The array is placed at the Tunka Advanced Instrument for cosmic rays and Gamma Astronomy (TAIGA) and detects the radio emission of air showers in the band of 30 to 80~MHz. During the last years it was shown that a sparse array such as Tunka-Rex is capable of reconstructing the parameters of the primary particle as accurate as the modern instruments. Based on these results we continue developing our data analysis. Our next goal is the reconstruction of cosmic-ray energy spectrum observed only by a radio instrument. Taking a step towards it, we develop a model of aperture of our instrument and test it against hybrid TAIGA observations and Monte-Carlo simulations. In the present work we give an overview of the current status and results for the last five years of operation of Tunka-Rex and discuss prospects of the cosmic-ray energy estimation with sparse radio arrays.Comment: Proceedings of E+CRS 201