Volume 46, № 6, 1994

We give a classification of maximal subalgebras of rankn?1 for the extended Poincaré algebra\(A\bar P (1.n)\), which is realized on the set of solutions of the d'Alembert equation\(\square u + \lambda u^k = 0\). These subalgebras are used for constructing anzatses that reduce this equation to differential equations with two invariant variables.

The Cauchy problem for the equation\(\partial u/\partial t = \mathcal{L}_x u = j(x) (u''_x )\) with positive essentially infinite-dimensional functionalsj(x) is studied in a properly chosen Banach space of functions on an infinite-dimensional separable real Hilbert space.

The well-known Neumann theorem on the isomorphism of standard wreath products is generalized to the wreath products of an arbitrary transitive permutation group and an abstract group.

A general approach is suggested for studying extremal properties of nondifferentiable convex functions on Euclidean combinatorial sets. On the basis of this approach, by solving the linear optimization problem on a set of combinations with repetitions, we obtain estimates of minimum values of convex and strongly convex objective functions in optimization problems on sets of combinations with repetitions and establish sufficient conditions for the existence of the corresponding minima.

The lattice of normal divisors of the Jonquere group over a field of characteristic zero is described.

A representation is obtained for a mixed sum of additive systems with values in a Banach ringX with identity and norm.

We give necessary conditions of equivalence of probability measures corresponding to generalized homogenous Gaussian fields. For the most part, the results are also new for standard homogenous fields and even in the one-dimensional case, i.e., for stationary processes.

We present the statement of control problems for singular distributed parabolic systems with discontinuous nonlinearties. Sufficient conditions for the existence of the optimal “control-state” pair are established under the assumption that the set of admissible “control-state” pairs is nonempty. The problem of existence of serniregular solutions is studied for the equation of state of a distributed system. It is not assumed that the nonlinearity of this equation increases sublinearly in the phase variable or possesses a bounded variation on any segment of the straight line.

For a system of nonlinear functional-differential equations with a linearly transformed argument, we establish the existence and uniqueness conditions for a solution bounded in the entire real axis and study the properties of this solution.

We prove theorem, in which basic conditions for the existence of nontrivial periodic solutions are formulated in terms of the properties of the elements of the matrix of a linear approximation to a system.

We give the complete description of the set of partial limits for a large class of sequences of weighted sums of independent random variables with triangular matrices of coefficients.

A method for constructing one-frequency solutions of nonlinear wave equations is suggested. This approach is based on a modified representation of asymptotic expansions by using special periodic Atebfunctions. This method makes it possible to obtain approximate solution of the problem under consideration without difficulty.

A criterion of harmonicity of functions in a Hilbert space is given in the case of weakened mutual dependence of the second derivatives.