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Given a normed plane, we consider the ranges of perimeters concerning some classes of triangles whose vertices are points on the unit sphere. The problems studied are essentially geometric. We try to collect several results that are scattered in different journals, comparing and improving some of them. Some illustrative examples are presented. Finally, some open problems are proposed.

Let X be a real Banach space. The rectangular constant $\\mu (X)$ and some generalisations of it, $\\mu _p(X)$ for $p \\geq 1$ , were introduced by Gastinel and Joly around half a century ago. In this paper we make precise some characterisations of inner product spaces by using $\\mu _p(X)$ , correcting some statements appearing in the literature, and extend to $\\mu _p(X)$ some characterisations of uniformly nonsquare spaces, known only for $\\mu (X)$ . We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning $\\mu (l_p)$ and $\\mu _p(l_p)$ .

A bstract In this paper, we report the measurement of the energy flow, the cross section and the average inelasticity of forward neutrons (+ antineutrons) produced in s = 13 TeV proton-proton collisions. These quantities are obtained from the inclusive differential production cross section, measured using the LHCf Arm2 detector at the CERN Large Hadron Collider. The measurements are performed in six pseudorapidity regions: three of them ( η > 10 . 75, 8 . 99 < η < 9 . 21 and 8 . 80 < η < 8 . 99), albeit with smaller acceptance and larger uncertainties, were already published in a previous work, whereas the remaining three (10 . 06 < η < 10 . 75, 9 . 65 < η < 10 . 06 and 8 . 65 < η < 8 . 80) are presented here for the first time. The analysis was carried out using a data set acquired in June 2015 with a corresponding integrated luminosity of 0 . 194 nb − 1 . Comparing the experimental measurements with the expectations of several hadronic interaction models used to simulate cosmic ray air showers, none of these generators resulted to have a satisfactory agreement in all the phase space selected for the analysis. The inclusive differential production cross section for η > 10 . 75 is not reproduced by any model, whereas the results still indicate a significant but less serious deviation at lower pseudorapidities. Depending on the pseudorapidity region, the generators showing the best overall agreement with data are either SIBYLL 2.3 or EPOS-LHC. Furthermore, apart from the most forward region, the derived energy flow and cross section distributions are best reproduced by EPOS-LHC. Finally, even if none of the models describe the elasticity distribution in a satisfactory way, the extracted average inelasticity is consistent with the QGSJET II-04 value, while most of the other generators give values that lie just outside the experimental uncertainties.

A bstract The forward η mesons production has been observed by the Large Hadron Collider forward (LHCf) experiment in proton-proton collision at s = 13 TeV. This paper presents the measurement of the inclusive production rate of η in p T < 1 . 1 GeV/c, expressed as a function of the Feynman- x variable. These results are compared with the predictions of several hadronic interaction models commonly used for the modelling of the air showers produced by ultra-high energy cosmic rays. This is both the first measurement of η mesons from LHCf and the first time a particle containing strange quarks has been observed in the forward region for high-energy collisions. These results will provide a powerful constraint on hadronic interaction models for the purpose of improving the understanding of the processes underlying the air showers produced in the Earth’s atmosphere by ultra-energetic cosmic rays.

A bstract In this paper, we report the measurement relative to the production of forward neutrons in proton-proton collisions at s = 13 TeV obtained using the LHCf Arm2 detector at the Large Hadron Collider. The results for the inclusive differential production cross section are presented as a function of energy in three different pseudorapidity regions: η > 10.76, 8.99 < η < 9.22 and 8.81 < η < 8.99. The analysis was performed using a data set acquired in June 2015 that corresponds to an integrated luminosity of 0.194 nb −1 . The measurements were compared with the predictions of several hadronic interaction models used to simulate air showers generated by Ultra High Energy Cosmic Rays. None of these generators showed good agreement with the data for all pseudorapidity intervals. For η > 10.76, no model is able to reproduce the observed peak structure at around 5 TeV and all models underestimate the total production cross section: among them, QGSJET II-04 shows the smallest deficit with respect to data for the whole energy range. For 8.99 < η < 9.22 and 8.81 < η < 8.99, the models having the best overall agreement with data are SIBYLL 2.3 and EPOS-LHC, respectively: in particular, in both regions SIBYLL 2.3 is able to reproduce the observed peak structure at around 1.5–2.5 TeV.

Multimessenger astrophysics is based on the detection, with the highest possible accuracy, of the cosmic radiation. During the last 20 years, the advent space-borne magnetic spectrometers in space (AMS-01, Pamela, AMS-02), able to measure the charged cosmic radiation separating matter from antimatter, and to provide accurate measurement of the rarest components of Cosmic Rays (CRs) to the highest possible energies, have become possible, together with the ultra-precise measurement of ordinary CRs. These developments started the era of precision Cosmic Ray physics providing access to a rich program of high-energy astrophysics addressing fundamental questions like matter-antimatter asymmetry, indirect detection for Dark Matter and the detailed study of origin, acceleration and propagation of CRs and their interactions with the interstellar medium.In this paper we address the above-mentioned scientific questions, in the context of a second generation, large acceptance, superconducting magnetic spectrometer proposed as mission in the context of the European Space Agency’s Voyage2050 long-term plan: the Antimatter Large Acceptance Detector In Orbit (ALADInO) would extend by about two orders of magnitude in energy and flux sensitivity the separation between charged particles/anti-particles, making it uniquely suited for addressing and potentially solving some of the most puzzling issues of modern cosmology.

In this paper, we report the measurement relative to the production of forward neutrons in proton-proton collisions at$$ \\sqrt{s}=13 $$s = 13 TeV obtained using the LHCf Arm2 detector at the Large Hadron Collider. The results for the inclusive differential production cross section are presented as a function of energy in three different pseudorapidity regions: η > 10.76, 8.99 < η < 9.22 and 8.81 < η < 8.99. The analysis was performed using a data set acquired in June 2015 that corresponds to an integrated luminosity of 0.194 nb −1 . The measurements were compared with the predictions of several hadronic interaction models used to simulate air showers generated by Ultra High Energy Cosmic Rays. None of these generators showed good agreement with the data for all pseudorapidity intervals. For η > 10.76, no model is able to reproduce the observed peak structure at around 5 TeV and all models underestimate the total production cross section: among them, QGSJET II-04 shows the smallest deficit with respect to data for the whole energy range. For 8.99 < η < 9.22 and 8.81 < η < 8.99, the models having the best overall agreement with data are SIBYLL 2.3 and EPOS-LHC, respectively: in particular, in both regions SIBYLL 2.3 is able to reproduce the observed peak structure at around 1.5–2.5 TeV.

The forward η mesons production has been observed by the Large Hadron Collider forward (LHCf) experiment in proton-proton collision at s = 13 TeV. This paper presents the measurement of the inclusive production rate of η in pT< 1.1 GeV/c, expressed as a function of the Feynman-x variable. These results are compared with the predictions of several hadronic interaction models commonly used for the modelling of the air showers produced by ultra-high energy cosmic rays. This is both the first measurement of η mesons from LHCf and the first time a particle containing strange quarks has been observed in the forward region for high-energy collisions. These results will provide a powerful constraint on hadronic interaction models for the purpose of improving the understanding of the processes underlying the air showers produced in the Earth’s atmosphere by ultra-energetic cosmic rays.

Given the good performances in terms of geometrical acceptance and energy resolution, calorimeters are the best suited detectors to measure high energy cosmic rays directly in space. However, in order to exploit this potential, the design of calorimeters must be carefully optimized to take into account all limitations related to space missions, due mainly to the mass of the experimental apparatus. CaloCube is a three years R&D project, approved and financed by INFN in 2014, aiming to optimize the design of a space-borne calorimeter by the use of a cubic, homogeneous and isotropic geometry. In order to maximize detector performances with respect to the total mass of the apparatus, comparative studies on different scintillating materials, different sizes of crystals and different spacings among them have been performed making use of Monte Carlo simulations. In parallel to this activity, several prototypes instrumented with CsI:Tl cubic crystals have been constructed and tested with particle beams (muons, electrons, protons and ions). Both simulations and prototypes showed that the CaloCube design leads to a good particle energy resolution (< 2% for electromagnetic showers, < 40% for hadronic showers) and a good effective geometric factor (> 3:5 m2 sr for electromagnetic showers, > 2:5 m2 sr for hadronic showers). Thanks to these performances, in 5 years of operation it would be possible to measure the ux of electrons+positrons up to some tens of TeV and the uxes of protons and nuclei up to some units of PeV/nucleon, hence extending these measurements by at least one order of magnitude in energy compared to the experiments currently operating in space.