Mathematics

2001, No. 2

M.I. Karpukhina, E.S. Kovalenko, I.S. Frolov

J-CONFIGURATIONS OF (0,1)-MATRICES AND THEIR INVARIANTS

One of the consequences of the well-known Holl's theorem on transversals is Ryser's theorem which asserts the existence of a (0,1)-matrix with given vectors of row sums and column sums. The paper deals with the (0,1)-matrix recovery problem by using more exact information, namely the lengths of blocks consisting of 1's in every line. The notion of matrix J-code is introduced and a family of matrices is examined for which the conditions of solvability are discovered and the algorithm of recovery is constructed. Also J-invariants, i.e. matrix functions which have equal values for matrices with the same J-code, are examined.

O.P. Filatov

THE PROOFS OF THE AVERAGING THEOREMS FOR DIFFERENTIAL INCLUSIONS

The new proofs for the averaging theorems of the system of differential inclusions with slow and rapid variables based on the theorem of continuous dependence of the solutions of the differential inclusions from their right hands and the initial conditions are presented.

V.A. Kouchmantseva

ONE MORE CONDITION FOR BANACH SPACE TO BE HIL'BERT

Contents of this article is a theorem, in which it is proved new sufficient condition for uniformly smooth and uniformly rotund Banach space to be Hil'bert space; the verification of this condition is rather easy than well-known parallelogram law. As a technique it is used the theory of spaces with semi-inner product by G.Lumer. In this note it is proved that if support mapping in uniformly smooth and uniformly rotund Banach space maps line segments to convex sets, then space is Hil'bert.

S.Y. Novikov

THE UNIQUENESS OF THE SYMMETRIC STRUCTURE OF THE SPACES

It is proved, that there is no rearrangement invariant space (RIS) E on [0,1], isomorphic to the space L¹[0,∞) ∩ L^∞[0,∞) or to the space L¹[0,∞)+L^∞[0,∞). Besides it's shown, that the inclusion map   L¹[0,∞)∩L^∞[0,∞)→E[0,∞) is strictly singular for each reflexive RIS E. The example given shows that the reflexivity condition is essential.

R.P. Uzbeckov

ON K-MONOTONIC COUPLES IN FINITE DIMENSIONAL SPACES

The problems related to the description of some K-monotonic couples in finite dimensional spaces are considered. Well-known the Calderon--Mityagin theorem is generalized on the Lorentz spaces. If w₁> ... w_n>0 then the finite dimensional Lorentz space λⁿ(w) is the space Rⁿ with the norm

We give the formula for the K-functional of the couple (λⁿ(w), l_∞ⁿ). We obtain a sufficient condition of the K-monotony of spaces of this class: if (λⁿ(w), l_∞ⁿ) is a precise K-monotonic couple then (w_n>0.)