# Volume 50, № 6, 1998

### On the existence and stability of periodic and almost periodic solutions of quasilinear equations with maxima

Bantsur N. R., Trofimchuk E. P.

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 747–754

We study the problem of existence of periodic and almost periodic solutions of the scalar equation *x*′ (*t*) = − δ*x*(*t*) + *p*max_{ u∈[t − h, t]} *x*(*u*) + *f*(*t*) where δ, *p* ∈ *R*, with a periodic (almost periodic) perturbation *f(t*). For these solutions, we establish conditions of global exponential stability and prove uniqueness theorems.

### Covariant derivatives of Jacobi fields on a manifold of nonpositive curvature

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 755–764

We obtain estimates of the covariant derivatives of Jacobi fields along a geodesic on a Riemannian manifold of negative curvature.

### Correlation inequalities for statistical models of lattice gas

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 765–773

We consider models of statistical mechanics of the type of lattice gas with attractive interaction of general kind. We propose a method for obtaining inequalities that connect multipoint correlation functions of different order. This method allows one, on the one hand, to strengthen similar inequalities, which can be obtained within the framework of the FKG method, and on the other hand, to obtain new inequalities. We introduce the notion of duality for models of lattice gas. We show that if, under the transformation *p* ⇒ 1 - *p*, the correlation inequalities for a model with attraction turn into the corresponding inequalities that are also satisfied, then the correlation functions of the dual model also satisfy the latter inequalities.

### Construction of a periodic system of moment equations

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 774–780

We construct a system of moment equations for a system of linear differential equations with periodic coefficients.

### Invariant symmetric restrictions of a self-adjoint operator. II

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 781–791

We give a criterion of invariance and symmetry of the restriction of an arbitrary unbounded self-adjoint operator in the space *L* _{2}(ℝ^{n}, *dx*) by using the introduced notion of support of an arbitrary vector and the notion of capacity of a subspace *N* ⊂ ℝ^{n}.

### Diametral theory of algebraic surfaces and geometric theory of invariants of groups generated by reflections. II

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 792–802

We systematically present the basic principles of the geometric theory of invariants of infinite groups generated by skew reflections with respect to hyperplanes in the real Euclidean space.

### Asymptotics of solutions of spectral problems of hydrodynamics in the neighborhood of angular points

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 803–811

At angular points on the boundary of a domain, we obtain an asymptotic expansion for the eigenfunctions of spectral problems that describe natural oscillations of an ideal liquid that partially fills a cavity in a solid body. We describe cases where the eigenfunctions have singularities at angular points.

### Representation of analytic functions by generalized series of exponents in unbounded convex domains

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 812–817

We establish conditions for the representation of functions *f(z*) analytic in unbounded convex domains *D* and continuous in \(\bar D\) via series of the form \(\sum\nolimits_{n = 1}^\infty {P_n \left( z \right)} e^{\lambda _n z} \) .

### Conditions of solvability of quasilinear periodic boundary-value problems for hyperbolic equations of the second order

Khoma G. P., Khoma N. H., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 818–821

On the basis of properties of the Vejvoda-Shtedry operator, we obtain solvability conditions for the 2π-periodic problem $$u_{tt} - u_{xx} = F\left[ {u,u_t } \right], u\left( {0,t} \right) = u\left( {\pi ,t} \right) = 0, u\left( {x,t + 2\pi } \right) = u\left( {x,t} \right)$$ .

### Dependence of separately continuous functions on n coordinates on products of compact sets

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 822–829

We investigate the problem of the dependence of separately continuous functions on n coordinates on products of spaces each of which is a topological product. In the case where *X* and *Y* are products of completely regular countably compact spaces, we establish necessary and sufficient conditions for such a dependence.

### On compactness and metrizability of nuclear spaces of manifolds

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 830–837

We prove a theorem on the sequential compactness of the so-called nuclear space of an arbitrary topological manifold.

### Information complexity of projection algorithms for the solution of Fredholm equations of the first kind. II

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 838–844

The optimal order of information complexity is found for certain classes of ill-posed problems.

### Asymptotic behavior of solutions of stochastic functional-differential equations with Poisson switchings

Gotinchan G. I., Yasinsky V. K.

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 845–861

We obtain conditions of global asymptotic stability of solutions of stochastic functional-differential equations with Poisson switchings.

### Balanced identity that describes *n*-ary group isotopes in the class of all *n*-ary quasigroups

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 862–864

We obtain a balanced identity that holds in a primitive *n*-ary quasigroup if and only if the polyadic quasigroup associated with the considered primitive *n*-ary quasigroup is a polyadic group isotope.