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p is a p-complete, nilpotent space and π1(BG. p )= G/Op(G). ... HBG = HBNG(D) = HB(NG(D)/Op′NG(D)) = HBDNG(D)/CG(D). Proof. See Swan [209].

it induces an algebra homomorphism. Φ : Hp,qd EndG(V(λp,q) Vd)op. (3.5). ... p,qd ). op thanks to Theorem 3.9. The laststatement follows from Corollary 3.8.

p(m)qkd (kA,kB) = Pr(KA = kA,KB = kB|M = m). (1). ... KA, kA, KB, kB: output keys for Alice and Bob.

1.4). 1.3. The analytic branches of eigenvalues and eigenvectors of A(t).According to the general analytic perturbation theory (see [Ka]), for |t| 6 t0there exist real-analytic functions

In spinors, if Ka = ιAoA. ′. then an α-plane distribution is defined by oA′. ,

Using the isometry z and the above identification of spinc bundles any Dirac op-. ... lim inf|k|. 1. |k|#{λ spec(PD,kA) : λ ε2k. } 1. 2π.

The geometri representation theory of these algebras has been studied in [Lu1, Lu2, Lu3 and fundamental re-sults have appeared in [HO, Op. ... The following proposition provides a graded He ke analogue of the results in [Ka,Proposition 1.20 and [Ka,

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JOURNAL OF MATHEMATICAL STUDYJ. Math. Study, Vol. x, No. x (201x), pp. 1-74. Astrophysical Dynamics and Cosmology. Tian Ma1, Shouhong Wang2,. 1 Department of Mathematics, Sichuan University, Chengdu, P. R. China.2 Department of Mathematics, Indiana