10,757 research outputs found

    Partition theorems and the Chinese remainder theorem

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    The famous partition theorem of Euler states that partitions of nn into distinct parts are equinumerous with partitions of nn into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of nn with all parts repeated at least once equals the number of partitions of nn where all parts must be even or congruent to 3(mod6)3 \pmod 6. These partition theorems were further extended by Glaisher, Andrews, Subbarao, Nyirenda and Mugwangwavari. In this paper, we utilize the Chinese remainder theorem to prove a comprehensive partition theorem that encompasses all existing partition theorems. We also give a natural generalization of Euler's theorem based on a special complete residue system. Furthermore, we establish interesting congruence connections between the partition function p(n)p(n) and related partition functions

    Kaon and pion parton distribution amplitudes to twist-three

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    We compute all kaon and pion parton distribution amplitudes (PDAs) to twist-three and find that only the pseudotensor PDA can reasonably be approximated by its conformal limit expression. At terrestrially accessible energy scales, the twist-two and pseudoscalar twist-three PDAs differ significantly from those functions commonly associated with their forms in QCD's conformal limit. In all amplitudes studied, SU(3) flavour-symmetry breaking is typically a 13% effect. This scale is determined by nonperturbative dynamics; namely, the current-quark-mass dependence of dynamical chiral symmetry breaking. The heavier-quark is favoured by this distortion, for example, support is shifted to the s-quark in the negative kaon. It appears, therefore, that at energy scales accessible with existing and foreseeable facilities, one may obtain reliable expectations for experimental outcomes by using these "strongly dressed" PDAs in formulae for hard exclusive processes. Following this procedure, any discrepancies between experiment and theory will be significantly smaller than those produced by using the conformal-limit PDAs. Moreover, the magnitude of any disagreement will either be a better estimate of higher-order, higher-twist effects or provide more realistic constraints on the Standard Model.Comment: 14 pages, 4 figures, 2 tables. arXiv admin note: text overlap with arXiv:1406.335

    Arithmetic Properties of Andrews' Singular Overpartitions

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    In a very recent work, G. E. Andrews defined the combinatorial objects which he called {\it singular overpartitions} with the goal of presenting a general theorem for overpartitions which is analogous to theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo 3 which followed from elementary generating function manipulations. In this work, we prove that Andrews' results modulo 3 are two examples of an infinite family of congruences modulo 3 which hold for that particular function. We also expand the consideration of such arithmetic properties to other functions which are part of Andrews' framework for singular overpartitions

    Congruences for rs(n)

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    AbstractLet rs(n) denote the number of representations of n as the sum of s squares of integers. In this note, we determine r2ks(n) modulo 2k+2 for kβ©Ύ1 and s odd. For general s, we also obtain a congruence for rs(n) modulo 2s. These extend Wagstaff's results (J. Number Theory 127 (2007) 326–329)

    Semileptonic Decays of BcB_c Meson to a P-Wave Charmonium State Ο‡c\chi_c or hch_c

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    The semileptonic decays of meson BcB_c to a P-wave charmonium state Ο‡c(3PJ)\chi_c(^3P_J) or hc(1P1)h_c(^1P_1) are computed. The results show that the decays are sizable so they are accessible in Tevatron and in LHC, especially, with the detectors LHCB and BTeV in the foreseeable future, and of them, the one to the 1P1^1P_1 charmonium state potentially offers us a novel window to see the unconfirmed hch_c particle. In addition, it is pointed out that since the two charmonium radiative decays Ο‡c(3P1,2)β†’J/ψ+Ξ³\chi_c(^3P_{1,2}) \to J/\psi+\gamma have sizable branching ratios, the cascade decays of the concerned decays and the charmonium radiative decays may affect the result of the observing the BcB_c meson through the semileptonic decays Bcβ†’J/ψ+l+Ξ½lB_{c}\to {J/\psi}+{l}+\nu_{l} substantially.Comment: 8 pages, 2 figure
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