Within the framework of the projective-geometric theory of differential equations developed by the author, a criterion for the rectifiability of a system of second-order differential equations is formulated in terms of the projective curvature tensor of the associated projective connection. Systems of second-order differential equations, with right-hand sides cubic in the first derivatives, admitting four-dimensional solvable symmetry groups of the Lie-Petrov type with a three-dimensional Abelian subgroup are found and studied. It is proved that there are only nine types of such systems ; for each of them, a rectifiability criterion is obtained and rectifying diffeomorphisms are presented. All systems , with a 4-dimensional solvable group of symmetries of Lie-Petrov type whose family of integral curves of is locally diffeomorphic to a family of straight lines have been found.
This article mainly gives a Cayley-type theorem for semitopological groups: Every semitopological group is algebraically and topologically embeddable into its homeomorphism group as a subgroup via a fixed embedding, where the homeomorphism group is given the topology of pointwise convergence.
structure of nonsingular matrices L E GL(v, 2) such that each row and each column of L and L-1 contains two or three 1's is described. It turns out that indecomposable matrices with this property exist only for v = 3, v = 5, and even v = 2k, k >= 2; moreover, in the last case, they can be represented in block form with 2 & times; 2 blocks. Up to independent permutations of rows and columns of the matrix L, the k blocks along the main diagonal consist of four 1's; k more equal blocks are located along the neighboring diagonal and are either the identity matrices (with two 1's on the diagonal) or matrices with one 1. The remaining blocks in L are zero (consist of four 0's). Encryption algorithms using such matrices are easy to implement, and they are free of the weaknesses of permutation matrices caused by the linear and difference methods of cryptographic analysis.
Success series in Bernoulli sequences of random variables are studied. A number of new results are obtained for the distributions of the number of successful trials before the emergence of various series and groups of successes.
