We complete the program started in [4] by proving that, in the weak coupling limit, the matrix elements, in the collective coherent vectors, of the Heisenberg evolved of an observable of a system coupled to a quasi-free reservoir through a laser type interaction, converge to the matrix elements of a quantum stochastic process satisfying a quantum Langevin equation driven by a quantum Brownian motion. Our results apply to an arbitrary quasi-free reservoir so, in particular, the finite temperature case is included.

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For a manifold with an S¹-action, we define generalized elliptic genera by using the orbit map and generalize Hirzebruch-Slodowy's formula in [9]. As a result, we have vanishing theorems of higher elliptic genera and higher twisted \hat{A}-genera. We also generalize elliptic genera of level N for stable almost complex manifolds and have a similar vanishing theorem.

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We define a notion of symmetric connections on subfactors and get a sufficient condition for a subfactor to have a symmetric connection. We also give a necessary and sufficient condition for Loi's invariant of a non-strongly outer automorphism of a subfactor to be trivial in the case with a symmetric connection. We apply this result to non-AFD SU(n)_k subfactors and construct orbifold subfactors of non-AFD SU(n)_k subfactors as well as the AFD case, as conjectured in our previous work. This generalizes constructions of Evans-Kawahigashi and Xu.

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Two classes of operator families, namely n-times integrated C-semigroups of hermitian and positive operators on Banach spaces, are studied. By using Gelfand transform and a theorem of Sinclair, we prove some interesting special properties of such C-semigroups. For instances, every hermitian nondegenerate n-times integrated C-semigroup on a reflexive space is the n-times integral of some hermitian C-semigroup with a densely defined generator; an exponentially bounded C-semigroup on L^p(μ)(1<p<∞) dominates C (a positive injective operator) if and only if its generator A is bounded, positive, and commutes with C; when C has dense range, the latter assertion is also true on L^p(μ) and C₀(Ω).

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We consider the Cauchy problem for the system of nonlinear wave equations
(*)   (∂_l²−Δ)u_l=F_l(u, u^', u^'')   in (0, ∞)×R², i=1, …, N
with initial data u_l(0, x)=εφ_l(x), (∂_lu_l)(0, x)=εψ_l(x), where F_l(i=1, …, N) are smooth functions of degree 3 near the origin (u, u^', u^'')=0, φ_l, ψ, ∈C₀^∞(R²) and ε is a small positive parameter. We assume that F_l(i=1, …, N) are independent of u_l^'' for any j≠i.
In the previous paper, the author showed the global existence of the small solution to the Cauchy problem (*) assuming that the cubic parts of the nonlinear terms satisfy Klainerman's null condition and that the nonlinear terms are independent of u_ju_ku_lu_m for any j, k, l, m=1, …, N. In this paper, we show the global existence without imposing further assumptions than the null condition on the cubic parts of the nonlinear terms.

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