Second-order necessary optimality conditions for domain optimization problems of the Neumann type

In this paper, we treat a domain optimization problem in which the boundary-value problem is a Neumann problem. In the case where the domain is in a three-dimensional Euclidean space, the first-order and the second-order necessary conditions which the optimal domain must satisfy are derived under a constraint which is the generalization of the requisite of constant volume.Portions of this paper were presented at the 13th IFIP Conference on System Modelling and Optimization, Tokyo, Japan, 1987.     

Meta-Hamiltonian groups with elementary commutant of rank 2

This paper is concerned with the structure of periodic meta-Hamiltonian groups with elementary commutant of rank 2 that have complemented Miller Moreno subgroups. It is proved that there exist four types of such groups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 168–175, February, 1990.The proof of this assertion will be given in another paper of the authors.     

A variational problem in the theory of green potential. I

For a condenser E in an open set D R ^p , p 3, the problem of the minimum of the Green energy in a class of charges, associated with E, is solved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 494–500, April, 1990.     

Properties of nonlinear waves in dissipative-dispersive media with instability

Wave processes in dissipative-dispersive media with instability described by a fourth-order nonlinear evolution equation are considered. Analytic solutions in the form of solitary and cnoidal waves are obtained. The existence of a critical value of the dispersion coefficient beyond which an initial disturbance (in particular, white noise) is transformed into a structure is demonstrated by numerical modeling.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 130–136, July–August, 1990.     

Quadratic spline interpolation on uniform meshes

The interpolation problem at uniform mesh points of a quadratic splines(x _i)=f _i,i=0, 1,...,N ands(x ₀)=f₀ is considered. It is known that s–f=O(h ³) and s–f=O(h ²), whereh is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent off other than mesh points wheres=f by assumption. Best error bounds for some compound formulae approximatingf _i andf _i ⁽³⁾ are also derived.     

Spectral expansions of some representations of Lie groups

We investigate representations of classical noncompact Lie groups induced by representations of the maximal parabolic subgroup. Formulas are obtained for 5O₀(2p, 2q)/U(p, q) and U(2p, 2q) Sp(p, q).Translated from Ukrainskii Mathematicheskii Zhurnal, Vol. 42, No. 1, pp. 128–131, January, 1990.     

Two-dimensional exponential model of the Euclidean field theory

It is proved that the two-dimensional exponential model of the field theory is trivial for ² > 8.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 469–477, April, 1990.     

A certain periodic control problem for differential equations with impulses in the space of bounded numerical sequences

One considers the differential equation dx/dt=f(t, x) with the impulse action ¦_t=t_i=H_i(t_i,x) in the space of bounded numerical sequences, where f(t, x), H_i(t, x) are T-periodic, countable-dimensional vector-valued functions, is a positive parameter. One gives conditions for the existence of a control (₁,₂) such that the solution of the equation dx/dt=f(t, x)–₁ with impulse action x¦_t=t_i=H_i(t_i,x)–₂ assuming for t= the value x=x₀, be T-periodic.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 271–275, February, 1990.     

Growth of analytic functions

For a function f analytic in a disk {z ¦ z ¦ <R }, 0 <R + , a criterion is obtained, in terms of the Taylor coefficients of the function, for the validity of the relation (r) ln⁺ M (r)dr < + , whereM (r)= max {¦ (f (z) ¦ ¦z ¦=r < R}, and is a function positive on [0, R) and satisfying certain conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 668–673, May, 1990.