Mathematics

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2006, No. 9


V.E.Voskresenskii

A CRITERION OF RATIONALITY OF AN ALGEBRAIC TORUS

The rationality of stably rational k-torus is proved. Therefore we have an algorithm enabling us to find algebraic tori which are rational over a basic field.

B.V. Loginov, O.V. Makeeva

THE SCHMIDT SPECTRAL PROBLEM WITH DISPLACEMENTS IN THE BOUNDARY CONDITIONS

In the paper the eigenvalue problem for the system of differential equations with displacements in the boundary conditions is considered to illustrate the E. Schmidt spectral problem. Eigenvalues and eigenfunctions for the direct and adjoint problems are determined.

S.P.Mishchenko, O.I. Cherevatenko

THE LEIBNIZ ALGEBRAS VARIETIES OF WEAK GROWTH

In the paper the varieties of the Leibniz algebras with the property that the codimension for some integer n restrict by the number 2 n/2 (so called the varieties of weakly growing codimension sequences) are studied. It is shown that such variety satisfies the identity

(x1x2)(x3x4)···(x2c+1x2c+2)≡0,

if the characteristic of a basic field is not equal to 2. We give also the equivalent condition for polynomial growth of varieties of the Leibniz algebras in terms of Young’s diagrams, if the characteristic of a basic field is equal to zero.