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    Energy dependent kinetic freeze-out temperature and transverse flow velocity in high energy collisions

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    Transverse momentum spectra of negative and positive pions produced at mid-(pseudo)rapidity in inelastic or non-single-diffractive proton-proton collisions and in central nucleus-nucleus collisions over an energy range from a few GeV to above 10 TeV are analyzed by a (two-component) blast-wave model with Boltzmann-Gibbs statistics and with Tsallis statistics respectively. The model results are in similarly well agreement with the experimental data measured by a few productive collaborations who work at the Heavy Ion Synchrotron (SIS), Super Proton Synchrotron (SPS), Relativistic Heavy Ion Collider (RHIC), and Large Hadron Collider (LHC), respectively. The energy dependent kinetic freeze-out temperature and transverse flow velocity are obtained and analyzed. Both the quantities have quick increase from the SIS to SPS, and slight increase or approximate invariability from the top RHIC to LHC. Around the energy bridge from the SPS to RHIC, the considered quantities in proton-proton collisions obtained by the blast-wave model with Boltzmann-Gibbs statistics show more complex energy dependent behavior comparing with the results in other three cases.Comment: 16 pages, 4 figures. The European Physical Journal A, accepted. arXiv admin note: text overlap with arXiv:1805.0334

    kkth power residue chains of global fields

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    In 1974, Vegh proved that if kk is a prime and mm a positive integer, there is an mm term permutation chain of kkth power residue for infinitely many primes [E.Vegh, kkth power residue chains, J.Number Theory, 9(1977), 179-181]. In fact, his proof showed that 1,2,22,...,2m11,2,2^2,...,2^{m-1} is an mm term permutation chain of kkth power residue for infinitely many primes. In this paper, we prove that for any "possible" mm term sequence r1,r2,...,rmr_1,r_2,...,r_m, there are infinitely many primes pp making it an mm term permutation chain of kkth power residue modulo pp, where kk is an arbitrary positive integer [See Theorem 1.2]. From our result, we see that Vegh's theorem holds for any positive integer kk, not only for prime numbers. In fact, we prove our result in more generality where the integer ring Z\Z is replaced by any SS-integer ring of global fields (i.e. algebraic number fields or algebraic function fields over finite fields).Comment: 4 page
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