Том 46, № 6, 1994

We give a classification of the maximal subalgebras of rank $n — 1$ for the extended Poincare algebra $A \widetilde{P}(1, n)$, which is realized on the set of solutions of the d'Alembeit equation $\Box u + \lambda u^k = 0$. These subalgebras are used for constructing the anzatses reducing this equation to differential equations with two invariant variables.

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

The well-known P. Neumann's theorem on the isomorphism of standard wreath products is generalized for the wreath products of an arbitrary transitive permutation group with an abstract group.

The general approach is suggested to the study of extremal properties of nondiffeientiable convex functions on Euclidean combinatorial sets. On the basis of this approach, by solving the linear optimization problem on the set of combinations with repetitions, we establish the estimates for the minimal values of convex and strongly convex objective functions in problems of optimization on the set of combinations with repetitions and the corresponding sufficient conditions for the existence of the minimum.

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

A representation is obtained for a mixed sum of additive systems with values on a Banach ring

Necessary conditions are given for the equivalence of the probability measures corresponding to the generalized homogeneous Gaussian fields. The results are mainly new also for the standard homogeneous fields as well, and even in the one-dimensional case, i.e., for stationary processes.

The critical points of the functionals $F:\; D \subset X \rightarrow \mathbb{R}$ defined on "nonlinear" sets $D$ in the topological vector spaces $X$ are studied. A construction of a $T$-derivative is suggested for these functionals and compared with to known constructions. The concept of a weak critical point is introduced and Coleman's principle is justified for $T$-differentiable functionals.

The statement of a control problem is given for singular distributed systems of parabolic type with discontinuous nonlinearities. The Sufficient conditions for the existence of an optimal pair "control-state" are obtained under the assumption that the set of admissible pairs "control-state" is nonempty. The problem of existence is studied for semiregular solutions of the equation of state of a distributed system. It is not assumed that the nonlinearity in the equation of state of a system has sublinear growth in the phase variable or 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 conditions of existence and uniqueness for a solution bounded on the entire real axis and study the properties of this solution.

In the space $\mathfrak{M}$ of bounded numeric sequences, we consider dichotomonly differential equation with a matrix projector. We prove that Green's function of this equation exists, and give the conditions, under which it is continuous and differentiable with respect to the parameters in the equation.

Theorems are proved, in which the main conditions for the existence of a nontrivial periodic solution are formulated in terms of the properties of the elements of a matrix of linear approximation to the system.

By using the methods of differential pulse-code modulation and "generalized" polygonal lines, we obtain almost exact estimates for the ε-entropy of the classes modeling signals of various types. The complexity of coding and recovering the functions from the classes under consideration is studied. A numerical solution is given for the problem of minimization of constants in the ordinal inequality for the ε-entmpy of the classes

We prove that the integral functionals with an integrand, which is a bounded function of a Wiener process on a cylinder, converge weakly to the proccss $w_1(\tau(t)), \tau(t) = \beta_1 t + (\beta_2 - \beta_1) {\rm mes} \{s;\;w_2(s) \geq 0, s

The complete description of the set of partial limits is given for a large class of sequences of weighted sums of independent random variables with a triangle coefficient matrix.

An approach to constructing one-frequency solutions of the nonlinear wave equation is given. The csscnce of the method is to write the asymptotic expansion in a modernized form by using the special periodic Ateb-functions. This method enables one to obtain an approximate solution to the problem under consideration without difficulties.

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

A quasicomplement $М$ ofasubspace $N$ of a Banach space $X$ is called strict if $M$ does not contain an infinite-dimensional subspace $M_1$, such that the linear manifold $N + M_1$, is closed. It is proved that if $X$ is separable, then $N$ always has a strict quasicomplement. We study the properties of the dense imbedding operator restricted to infinite-dimensional closed subspaces of the space, where it is defined.