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    The longitudinal and transverse distributions of the pion wavefunction from the present experimental data on the pion-photon transition form factor

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    It is noted that the low-energy behavior of the pion-photon transition form factor Fπγ(Q2)F_{\pi\gamma}(Q^2) is sensitive to the transverse distribution of the pion wavefunction, and its high-energy behavior is sensitive to the longitudinal one. Thus a careful study on Fπγ(Q2)F_{\pi\gamma}(Q^2) can provide helpful information on the pion wavefunction precisely. In this paper, we present a combined analysis of the data on Fπγ(Q2)F_{\pi\gamma}(Q^2) reported by the CELLO, the CLEO, the BABAR and the BELLE collaborations. It is performed by using the method of least squares. By using the combined measurements of BELLE and CLEO Collaborations, the pion wavefunction longitudinal and transverse behavior can be fixed to a certain degree, i.e. we obtain β[0.691,0.757]GeV\beta \in [0.691,0.757] \rm GeV and B[0.00,0.235]B \in [0.00,0.235] for Pχ290%P_{\chi^2} \geq 90\%, where β\beta and BB are two parameters of a convenient pion wavefunction model whose distribution amplitude can mimic the various longitudinal behavior under proper choice of parameters. We observe that the CELLO, CLEO and BELLE data are consistent with each other, all of which prefers the asymptotic-like distribution amplitude; while the BABAR data prefers a more broad distribution amplitude, such as the CZ-like one.Comment: 7 pages, 10 figure

    On uniqueness of heat flow of harmonic maps

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    In this paper, we establish the uniqueness of heat flow of harmonic maps into (N, h) that have sufficiently small renormalized energies, provided that N is either a unit sphere Sk1S^{k-1} or a compact Riemannian homogeneous manifold without boundary. For such a class of solutions, we also establish the convexity property of the Dirichlet energy for tt0>0t\ge t_0>0 and the unique limit property at time infinity. As a corollary, the uniqueness is shown for heat flow of harmonic maps into any compact Riemannian manifold N without boundary whose gradients belong to LtqLxlL^q_t L^l_x for q>2q>2 and l>nl>n satisfying the Serrin's condition.Comment: 24 pages. Two errors of proof of lemma 2.3 have been fixe