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We consider a kinetic model of self-propelled particles with alignment interaction and with precession about the alignment direction. We derive a hydrodynamic system for the local density and velocity orientation of the particles. The system consists of the conservative equation for the local density and a non-conservative equation for the orientation. First, we assume that the alignment interaction is purely local and derive a first order system. However, we show that this system may lose its hyperbolicity. Under the assumption of weakly non-local interaction, we derive diffusive corrections to the first order system which lead to the combination of a heat flow of the harmonic map and Landau-Lifschitz-Gilbert dynamics. In the particular case of zero self-propelling speed, the resulting model reduces to the phenomenological Landau-Lifschitz-Gilbert equations. Therefore the present theory provides a kinetic formulation of classical micromagnetization models and spin dynamics.

Based on the proof of Labastida-Mari {} o-Ooguri-Vafa conjecture\cite {lmov}, we derive an infinite product formula for Chern-Simons partition functions, the generating function of quantum $\fsl_N $ invariants. Some symmetry properties of the infinite product will also be discussed.

Kawakubo and Uchida showed that, if a closed oriented 4k-dimensional manifold M admits a semi-free circle action such that the dimension of the fixed point set is less than 2k, then the signature of M vanishes. In this note, by using G-signature theorem and the rigidity of the signature operator, we generalize this result to more general circle actions. Combining the same idea with the remarkable WittenTaubesBott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of ConnerFloyd, LandweberStong and HirzebruchSlodowy.

0. Introduction. Let Mn be a n-dimensional minimally immersed submanifold of M, Q> 1. Throughout this paper M is taken to be one of the simply connected space forms with curvature 1, 0, or-1, ie Mfn+ f sn+ f, Rn+ f, or Hn+ f. Given a point p EM, let rp (x) be the dis-tance function on M, we denote the restriction of rp to M as the extrinsic distance function on M. For any a> 0, we define the extrinsic ball cen-tered at p with radius a by