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In this paper we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct sum of null-filiform ideals, and a onedimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra, as well.2010 Mathematics Subject Classification. 17A32, 17A65, 17B30.

The structural constants of an evolution algebra are given by a quadratic matrix. In this work we establish an equivalence between nil, right nilpotent evolution algebras and evolution algebras defined by upper triangular matrices. The classification of 2-dimensional complex evolution algebras is obtained. For an evolution algebra with a special form of the matrix, we describe all its isomorphisms and their compositions. We construct an algorithm running under Mathematica which decides if two finite dimensional evolution algebras are isomorphic.

Abstract. In this work we investigate the derivations of n−dimensional complex evolution algebras, depending on the rank of the appropriate matrices. For evolution algebra with non-singular matrices we prove that the space of derivations is zero. The spaces of derivations for evolution algebras with matrices of rank n − 1 are described.

Abstract. The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras the criterium of nilpotency is established in terms of the properties of corresponding matrices. Moreover, it is proved that for nilpotent n-dimensional complex evolution algebras the possible maximal nilpotency index is 1 + 2 n−1 .

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We provide a solvability criteria for a depressed cubic equation in domains Z * p , Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the depressed cubic equation in domains Z * p , Zp, Qp are provided. Since Fp ⊂ Qp, we generalize J.-P. Serre's [27] and Z.H.Sun's [28,30] results concerning with depressed cubic equations over the finite field Fp. Finally, all depressed cubic equations, for which the Cardano method could be applied, are described and the p−adic Cardano formula is provided for those cubic equations. Mathematics Subject Classification: 11Sxx

In this paper we show that the method for describing solvable Lie algebras with given nilradical by means of non-nilpotent outer derivations of the nilradical is also applicable to the case of Leibniz algebras. Using this method we extend the classification of solvable Lie algebras with naturally graded filiform Lie algebra to the case of Leibniz algebras. Namely, the classification of solvable Leibniz algebras whose nilradical is a naturally graded filiform Leibniz algebra is obtained.