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We study Kuzmin's conjecture on the index of nilpotency for the variety N il 5 of associative nil-algebras of degree 5. Due to Vaughan-Lee  the problem is reduced to that for k-generator N il 5-superalgebras, where k ≤ 5. We confirm Kuzmin's conjecture for 2-generator superalgebras proving that they are nilpotent of degree 15.
We extend the Jacobson’s Coordinatization theorem to Jordan superalgebras. Using it we classify Jordan bimodules over superalgebras of types Q(n) and JP (n), n ≥ 3. Then we use the Tits-Kantor-Koecher construction and representation theory of Lie superalgebras to treat the remaining case Q(2).
We obtain classification of the irreducible bimodules over the Jordan superalgebra Kan(n), the Kantor double of the Grassmann Poisson superalgebra G n on n odd generators, for all n ≥ 2 and an algebraically closed field of characteristic = 2. This generalizes the corresponding result of C.Martínez and E.Zelmanov announced in [MZ2] for n > 4 and a field of… (More)
Recently J.A.Anquela, T.Cortés, and H.Petersson  proved that for elements x, y in a non-degenerate Jordan algebra J, the relation x • y = 0 implies that the U-operators of x and y commute: U x U y = U y U x. We show that the result may be not true without the assumption on non-degeneracity of J. We give also a more simple proof of the mentioned result in… (More)
We prove the unique solvability of a mixed boundary value problem for the p-Laplace operator by means of variational methods. Using the obtained results, we construct an iterative procedure for solving the Cauchy problem for the p-Laplace operator.