In this paper, we show rigorously that there exists the ferrimagnetic long-range order in the ground state of the positive-U Hubbard model at half 6lling on some bipartite lattices. When NQ ) NQ (N~and Ns are the total site numbers of two sublattices A and B), except for the ferromagnetism which was found by Lieb [Phys. Rev. Lett. 62, 1201(1989], there also exists the antiferromagnetic long-range order in the ground state. This result only requires U )0 and is independent of the dimension of the lattices. PACS numbers: 75.10. -Lp, 05.30.Fk, 75.50.GgThe existence of antiferromagnetic long-range order in an itinerant electron system has been an open problem for a long time. Four years ago, Lieb showed that the unique ground state of the Hubbard model on some bipartite lattices at half filling has the total spin S = ]NA -Np]/2 [1]. When NA )Nz, it is nonzero macroscopically. However, in the large-U case, this model is very close to an antiferromagnetic Heisenberg model. It is believed that the ground state should favor the antiferromagnetism, or more accurately, the ferrimagnetism. Except for the nonzero spin, the ferrimagnetism also possesses the antiferromagnetic long-range order. Thus whether there exists such a long-range order is the subject of this paper.After some preliminary definitions and introduction, we present a theorem on two-point spin correlation function in the Hubbard model. Then incorporating Lieb's theorem [1], we prove that when NA )NB [2] and at halffilling, i.e. , N, = NA+N~, the ground state possesses the ferrimagnetic long-range order. To our knowledge, this is the first example that the ferrimagnetic long-range order exists exactly in an itinerant electron system.The positive-U Hubbard model on a finite bipartite lattice A is defined by t). C, '. C-+-]Ul).(n'i--, ')(n'1 --, '), (1) (~i)w here C; and C, . are the annihilation and creation operators of electron with spin cr (=t', $) at site i, respectively. (ij) means all pairs of the nearest neighbor sites and i and j belong to two sublattices. This model possesses SO(4) symmetry [3] and the global ground state must belong to the subspace, Nt = Nt (Nt and Nt are the total numbers of electron with spin-up and spin-down, respectively, and both are the good quantum number of the system). Therefore we limit our discussion in this subspace.The raising and lowering spin operators are S, + = C, . iC,t, S, = C, . tC, T, and the z component is (2) m(q) = l $ e(~)s+~) c(1)s, )), where (. *(i) = 1 when i 6 A, and -1 when i c B and the angular brackets denote the ground state expectation. The existence of ferromagnetic (antiferromagnetic) long-range order requires that m(0) [m(q)] has order of NA (NA = NA+N~), i.e. , m(0) = 0(Np). The ferrimagnetism requires that both m(0) and m(Q) have order of NA. Now we present Theorem I. Theorem I. -In the positive-U Hubbard model on the bipartite lattice A and at half-filling,

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We give a more complete calculation of b -> sj decay, including leading log QCD corrections from m t to Mw in addition to corrections from Mw to nib. We have included the full set of dimension-6 operators and corrected numerical mistakes of anomalous dimensions in a previous paper [14]. Compared with the calculations without QCD running from m t to Mw, the inclusive decay rate is found to be enhanced. At m t = 150 GeV, it results in a 12% enhancement, and for m t = 250 GeV, 15% is found. The total QCD effect makes an enhanced factor of 4.2 at m t = 150 GeV, and 3.2 for m t = 250 GeV.PACS number(s): 13.25. Hw, 12.38.Bx, 13.40.Hq