A Guide to the Russian Academy of Sciences - University of Texas ...

It have been shown that metric projection on the every non-empty convex closed set of the
B-space X is one-valued and continuous iff X is a strictly convex Efimoff-Stechkin
space. It has been also proved that every Chebyshev set of uniform convex B-space
is a connected one (L. P. Vlasov).
It has been shown that Chebyshev set of the uniform convex space with the Frechet
differentiable norm is convex if cardinal number of the discontinuity set for the
metric projections is at most continuum (V. S. Balaganskii).
The best on every parameter estimates have been established for the numerical characteristic
of stability of the metric d-projection on the convex set M of linear metric space
with respect to the data error namely of set M and the element to be projected. The
criterion of the uniform strong unicity of the best Chebyshev approximation and
solution of the more general extreme problem has been founded (A. V. Marinov).
The questions of existence of Chebyshev system on the compacts, connected with
generalization of Mairhuber theorem had been studied (V. A. Koscheev).
The numerical methods of approximation are applicable in the various branches of science
concerned with the complicated functions and bulky massif of the test data known
with an error to compress the great massives of an test data, to recover the
information on incomplete data, to smooth out the inexact experimental information.
Methods of approximation have been developed in the Institute for one- and severalvariable
functions in various metrics by means of generalized polynomials and
sums of exponents (V. P. Kondratiev), rational fractions (L. V. Petrak), splines
(A. A. Sazanov, N. L. Patzko) etc. Software for mentioned methods was worked
out. Proposed methods and software allowed us to solve many problems related
with the development of a new technique: some questions, connected with the
motion control of a material points, approximation of a bulky volumes of a
metereological information, the hardness calculus of rubber-metal details,
elaboration of non-destructive testing of some details, diagnostic of the blood
current disturbance in the pancreas, optimization of the geometry and characteristics
of the hybrid reflector antennas and elaboration of the algorithms for the control of
such system in particular of systems with only double-phase control.
The scientific research have been headed by V. I. Berdyshev, N. I. Chernykh, S. B.
Stechkin, Iurii N. Subbotin. One of the problem has been developed jointly with
the Steklov Mathematical Institute.
Ill-posed Problems Department
The Department of ill-posed problems of analysis and application was organized in
November 1990 (formerly it was a section (underdepartment) of the Derpartment of
applied problems). Its chief is Professor V. Vasin.
The principle field of the Derpartment research is the investigation on the theory and
methods of ill-posed and inverse problems solutions.
The optimality of some classes of regularizing algorithms has been proved for linear and
non-linear operator equations as well as for the problem of evaluating of values for
unbounded operator.
The general scheme of discrete approximation for the extreme problems has been
suggested. Necessary and sufficient conditions for convergence of the finite-
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