Volume 2, Issue 1

The aim of this paper is to find the class of continuous pointwise transformations (as general as possible) in the framework of which Kummer's transformation z ( t ) = g ( t ) y ( h ( t )) represents the most general pointwise transformation converting every linear homogeneous differential equation of the n th order into an equation of the same type. Further, some forms of these equations having certain subspaces of solutions aer cobstructed.

It is shown that the right-sided, left-sided, and symmetric maximal functions of any measurable function can be integrable only simultaneously. The analogous statement is proved for the ergodic maximal functions.

The periodic boundary-value problem for systems of second-order ordinary nonlinear differential equations is considered. Sufficient conditions for the existence and uniqueness of a solution are established.

The k th root taken of the bosonic phase operator is considered. This leads to the extension of the Hilbert space. In the case k = 2 an ordinary fermionic extension arises. Particles whose statistics depends on k are introduced for other values of k .

Some properties of Borel measures with separable supports are considered. In particular, it is proved that any σ -finite Borel measure on a Suslin line has a separable supports and from this fact it is deduced, using the continuum hypothesis, that any Suslin line contains a Luzin subspace with the cardinality of the continuum.

It is shown that if Π n , n > 2, are Chogoshvili's cohomotopy functors [Chogoshvili, Bull. Acad. Sci. Georgian SSR 97: 273-276, 1980, Khazhomia, Trudy Tbiliss. Mat. Inst. Razmadze 91: 81-87, 1988, Khazhomia, Georgian Math. J. 1: 151-171, 1994], then 1) the isomorphism Π n ( X 1 × X 2 ) ≈ Π n ( X 1 ) ⊕ Π n ( X 2 ) holds for topological spaces X 1 and X 2 ; 2) the relations rank Π n ( X ) = rank π n ( X ), Tor Π n +1 ( X ) ≈ Tor π n ( X ) hold under certain restrictions for the space X .

Conditions are found upon satisfaction of which the differential equation x ( n ) ( t ) – λ x ( n ) ( t – σ ) + f ( t , x ( g ( t ))) = 0 has solutions which are asymptotically equivalent to the solutions of the equation x ( n ) ( t ) – λ x ( n ) ( t – σ ) = 0.

The sufficient conditions are established under which the second-order linear differential equation is conjugate.

Two-point boundary-value problems are investigated by the method of barriers for ordinary differential equations of the second and the third order.