In the paper [Som90], Somekawa conjectures that his Milnor K-group K(k,G₁,…,G_r) attached to semi-abelian varieties G₁,…,G_r over a field k is isomorphic to Ext_{$¥mathcal{M}$k}^r($¥mathbb{Z}$,G₁[-1] ⊗ … ⊗ G_r[-1]) where $¥mathcal{M}$_k is a certain category of motives over k. The purpose of this note is to prove this conjecture, when the varieties G_i are Jacobians of smooth curves over a perfect field and we take $¥mathcal{M}$_k as Voevodsky's category of motives DM_-^eff(k).

Let G be a finite group, and let p be a prime number. It might happen that the p-Sylow normalizer N_G(P), P ∈ Syl_p(G), of G is p-nilpotent, but G will not be p-nilpotent (see Example 1.1). However, under certain hypothesis on the structure of the Sylow p-subgroup P of G, this phenomenon cannot occur, e.g., by J. Tate's p-nilpotency criterion this is the case if P is a Swan group in the sense of H-W. Henn and S. Priddy. In this note we show that if P does not contain subgroups of a certain isomorphism type Y_p(m) — in which case we call the p-group P slim — the previously mentioned phenomenon will not occur provided p is odd. For p = 2 the same remains true if P is D₈-free (see Main Theorem).

We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group U($¥mathcal{H}$) in a Hilbert space $¥mathcal{H}$ with U($¥mathcal{H}$) equipped with the strong operator topology. More precisely, for any strongly closed subgroup G of the unitary group U($¥mathfrak{M}$) in a finite von Neumann algebra $¥mathfrak{M}$, we show that the set of all generators of strongly continuous one-parameter subgroups of G forms a complete topological Lie algebra with respect to the strong resolvent topology. We also characterize the algebra $¥overline{¥mathfrak{M}}$ of all densely defined closed operators affiliated with $¥mathfrak{M}$ from the viewpoint of a tensor category.

Using strong commuting self-adjoint operators in the Minkowski space, we showed that the operator concerning a neutral particle with an anomalous magnetic moment is related to that of a free particle by a non-unitary transformation.

We consider the limit of the iteration of a map z ↦ m(z) from a complex domain D to D. For two kinds of maps m, we show that each iteration mⁿ(z) of m(z) converges for any z ∈ D as n → ∞ and that this limit is expressed by the hypergeometric function. These are analogs of the expression of the arithmetic-geometric mean by the Gauss hypergeometric function.