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The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.  相似文献   
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The aqueous vanadium(III) (V(III)) speciation chemistry of two dipicolinate-type complexes and the insulin-enhancing effects of V-dipicolinate (V-dipic) complexes in three different oxidation states (V(III), V(IV), and V(V)) have been studied in a chronic animal model system. The characterization of the V(III) species was carried out at low ionic strength to reflect physiological conditions and required an evaluation of the hydrolysis of V(III) at 0.20 M KCl. The aqueous V(III)-dipic and V(III)-dipic-OH systems were characterized, and complexes were observed from pH 2 to 7 at 0.2 M KCl. The V(III)-dipic system forms stable 1:2 complexes, whereas the V(III)-dipic-OH system forms stable 1:1 complexes. A comparison of these complexes with the V-pic system demonstrates that a second ligand has lower affinity for the V(III), presumably reflecting bidentate coordination of the second dipic(2)(-) to the V(III). The thermodynamic stability of the [V(III)(dipic)(2)](-) complex was compared to the stability of the corresponding V(IV) and V(V) complexes, and surprisingly, the V(III) complexes were found to be more stable than anticipated. Oral administration of three V-dipicolinate compounds in different oxidation states {H[V(III)(dipic)(2)H(2)O].3H(2)O, [V(IV)Odipic(H(2)O)(2)].2H(2)O, and NH(4)[V(V)O(2)dipic]} and the positive control, VOSO(4), significantly lowered diabetic hyperglycemia in rats with streptozotocin-induced diabetes. The diabetic animals treated with the V(III)- or V(IV)-dipic complexes had blood glucose levels that were statistically different from those of the diabetic group. The animals treated with the V(V)-dipic complex had the lowest blood glucose levels of the treated diabetic animals, which were statistically different from those of the diabetic group at all time points. Among the diabetic animals, complexation to dipic increased the serum levels of V after the administration of the V(V) and V(IV) complexes but not after the administration of the V(III) complex when data are normalized to the ingested dose of V. Because V compounds differing only in oxidation state have different biological properties, it is implied that redox processes must be important factors for the biological action of V compounds. We observe that the V(V)-dipic complex is the most effective insulin-enhancing agent, in contrast to previous studies in which the V(IV)-maltol complex is the most effective. We conclude that the effectiveness of complexed V is both ligand and oxidation state dependent.  相似文献   
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