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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 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 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 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.

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.

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.

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.

The Large Hadron Collider forward and the Relativistic Heavy Ion Collider forward experiments measured forward particles produced in high-energy hadron collisions at the LHC and RHIC. Using compact calorimeters neutral particles produced in pseudorapidities η >8.4 and η >6.0 are observed by the respective experiments. Because the collision energies ranging from 0.51 TeV to 13 TeV correspond to the cosmic-ray equivalent energies of 10 14 to 10 17 eV, the measurements are important to understand the hadronic interaction relevant to extensive air shower measurements. This paper reviews recent results of LHCf and initial performance of RHICf that took data in the 2017 RHIC operation.

Future space experiments dedicated to the observation of high-energy gamma and cosmic rays will increasingly rely on a highly performing calorimetry apparatus, and their physics performance will be primarily determined by the geometrical dimensions and the energy resolution of the calorimeter deployed. Thus it is extremely important to optimize its geometrical acceptance, the granularity, and its absorption depth for the measurement of the particle energy with respect to the total mass of the apparatus which is the most important constraint for a space launch. The proposed design tries to satisfy these criteria while staying within a total mass budget of about 1.6 tons. Calocube is a homogeneous calorimeter instrumented with Cesium iodide (CsI) crystals, whose geometry is cubic and isotropic, so as to detect particles arriving from every direction in space, thus maximizing the acceptance; granularity is obtained by filling the cubic volume with small cubic CsI crystals. The total radiation length in any direction is more than adequate for optimal electromagnetic particle identification and energy measurement, whilst the interaction length is at least suficient to allow a precise reconstruction of hadronic showers. Optimal values for the size of the crystals and spacing among them have been studied. The design forms the basis of a three-year R&D activity which has been approved and financed by INFN. An overall description of the system, as well as results from preliminary tests on particle beams will be described.