Commuting Flows and Conservation Laws for Noncommutative Lax Hierarchies

We discuss commuting flows and conservation laws for Lax hierarchies on noncommutative spaces in the framework of the Sato theory. On commutative spaces, the Sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the Lax hierarchies in terms of pseudo-differential operators. Noncommutative extension of the Sato theory has been already studied by the author and Kouichi Toda, and the existence of various noncommutative Lax hierarchies are guaranteed. In the present paper, we present conservation laws for the noncommutative Lax hierarchies with both space-space and space-time noncommutativities and prove the existence of infinite number of conserved densities. We also give the explicit representations of them in terms of Lax operators. Our results include noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera, modified KdV equations and so on.Comment: 22 pages, LaTeX, v2: typos corrected, references added, version to appear in JM

Pourines. XLV. : Syntheses and Cytokinin Activities of the Cis Isomers of (1\u27R)-1\u27-Methylzeatin and Its 9-β-D-Ribofuranoside

(1\u27R)-1\u27-Methyl-cis-zeatin (2b) and its 9-β-D-ribofuranoside (4b) have been synthesized for the first time from D-alanine (5) in 7 steps. The new cis-zeatin derivatives 2b and 4b, together with known cis-zeatin (2a), were tested for cytokinin activity in the tobacco callus and the lettuce seed germination bioassays. In both bioassays, the cytokinin activity was found to follow the order 2a>2b>4b, indicating that 2b and 4b were less active than the corresponding trans isomers 1b and 3b (natural cytokinins from Pseudomonas syringae pv savastanoi), respectively

Spin Injection into a Graphene Thin Film at Room Temperature

We demonstrate spin injection into a graphene thin film with high reliability by using non-local magnetoresistance (MR) measurements, in which the electric current path is completely separated from the spin current path. Using these non-local measurements, an obvious MR effect was observed at room temperature; and the MR effect was ascribed to magnetization reversal of ferromagnetic electrodes. This result is a direct demonstration of spin injection into a graphene thin film. Furthermore, this is the first report of spin injection into molecules at room temperature.Comment: 12 pages, 3 figure