Volume 49, № 12, 1997

For the generalized many-dimensional Lavrent’ev-Bitsadze equation, we prove the unique solvability of the Dirichlet and Tricomi problems. We also establish the existence and uniqueness of a solution of the Dirichlet problem in the hyperbolic part of a mixed domain.

The relation between the physical theory and its mathematical formalism is shown.

We use energy methods to prove the existence and uniqueness of solutions of the Dirichlet problem for an elliptic nonlinear second-order equation of divergence form with a superlinear tem [i.e., g(x, u)=v(x) a(x)⋎u⋎ ^p−1u,p>1] in unbounded domains. Degeneracy in the ellipticity condition is allowed. Coefficients a _i,j(x,r) may be discontinuous with respect to the variable r.

We establish conditions under which, for a Dirichlet series $F(z) = \sum_{n = 1}^{∞} d n \exp(λ_n z)$, the inequality $⋎F(x)⋎ ≤ y(x),\quad x ≥ x_0$, implies the relation $\sum_{n = 1}^{∞} |d_n \exp(λ_n z)| ⪯ γ((1 + o(1))x)$ as $x → +∞$, where $γ$ is a nondecreasing function on $(−∞,+∞)$.

An asymptotic solution of a system of inegro-differential equations is constructed for the case where turning points are present.

We study parabolic equations of the divergent form with degeneration. We have obtained an estimate for the maximum of modulus of generalized solutions of the first boundary-value problem with a zero on the parabolic boundary.

We consider the problem of coding of parametrically defined vector functions by continuous functionals which act on their coordinate functions. We obtain the optimal method for coding some class of vector functions. In the linear case, we show the relation between this method and the informativeness of functionals with respect to the class of coordinate functions.

We obtain the characteristic for radical algebras subgroups of whose adjoint groups are subalgebras. In particular, we prove that the algebras of this sort are nilpotent with nilpotent length at most three. We give the complete classification of those algebras under consideration which are generated by two elements.

We study the structure of the distribution of a complex-valued random variable ξ = Σa _k ξ _k , where ξ _k are independent complex-valued random variables with discrete distribution and a _k are terms of an absolutely convergent series. We establish a criterion of discreteness and sufficient conditions for singularity of the distribution of ξ and investigate the fractal properties of the spectrum.

We investigate the behavior of a diserete dynamical system in a neighborhood of a quasiperiodic trajeetory for the case of an infinite-dimensional Banach space We find conditions sufficient for the system considered to reduce, in such a neighborhood, to a system with quasiperiodic coefficients.

We consider the problem of conditions for the existence of multiple singular integrals of a certain class at inner and boundary points of a domain. We obtain the quadrature and cubature formulas for calculating multiple singular integrals and present the corresponding estimates for the formulas.

The problem with unknown boundaries for a first-order semilinear hyperbolic system is studied in the case where the curve of definition of the initial conditions degenerates to a point. An existence and uniqueness theorem for a classical solution of the problem is proved for small t.

We study a periodie boundary-value problem for the quasilinear equation u _tt − u _xx = F[u, u _t , u _x ]. We find conditions under which a theorem on the uniqueness of a solution is true.

We obtain exact (unimprovable) estimates for the rate of convergence of the s-step method of steepest descent for finding the least (greatest) eigenvalue of a linear bounded self-adjoint operator in a Hilbert space.

Lower estimates of the Kolmogorov widths are obtained for certain classes of infinitely differentiable periodic functions in the metrics of C and L. For many important cases, these estimates coincide with the values of the best approximations of convolution classes by trigonometric polynomials calculated by Nagy, and, hence, they are exact.

We formulate a convenient general method for construating a complets set of comformal-like series of conservation laws of the n th order. As examples, we give all conformal-like series which are generated by symmetric tensors of the third and fourth ranks.

We study the boundary value problem for the quasilinear equation u _u − u_xx=F[u, u_t], u(x, 0)= u(x, π)=0, u(x+w, t)=u(x, t), x ε ®, t ε [0, π], and establish conditions under which a theorem on the uniqueness of a smooth solution is true.

We establish necesary and sufficient conditions for the exponential diehetemy with matrix projeseters of a countable system of differential equations with quasiperiodic coefficients.