Volume 3, Issue 1

We discuss the commutativity of certain rings with unity 1 and one-sided s -unital rings under each of the following conditions: x r [ x s , y ] = ±[ x, y t ] x n , x r [ x s , y ] = ± x n [ x, y t ], x r [ x s , y ] = ±[ x, y t ] y m , and x r [ x s , y ] = ± y m [ x, y t ], where r, n , and m are non-negative integers and t > 1, s are positive integers such that either s, t are relatively prime or s [ x, y ] = 0 implies [ x, y ] = 0. Further, we improve the result of [Jacobson, Structure of rings, Colloq. Publ., 1964, Theorem 3] and reprove several recent results.

Partial averaging for impulsive differential equations with supremum is justified. The proposed averaging schemes allow one to simplify considerably the equations considered.

For a fibration with the fiber K ( π, n )-space, the algebraic model as a twisted tensor product of chains of the base with standard chains of K ( π, n )-complex is given which preserves multiplicative structure as well. In terms of this model the action of the n -cohomology of the base with coefficients in π on the homology of fibration is described.

The uniqueness theorem for the one-sided maximal operator has been proved.

The question of the correct formulation of a Darboux type non-characteristic spatial problem for the wave equation is investigated. The correct solvability of the problem is proved in the Sobolev space for surfaces of the temporal type on which Darboux type boundary conditions are given.

Littlewood–Paley operators defined on a new kind of generalized Lipschitz spaces are studied. It is proved that the image of a function under the action of these operators is either equal to infinity almost everywhere or is in , where – n < α < 1 and 1 < p < ∞.

The problem is posed and solved whether the conditions and sup θ ∈[0, π /2) ∫ { M 2, θ (ƒ) > 1} M 2, θ (ƒ) < ∞ are equivalent for functions (where M 2, θ denotes the strong maximal operator corresponding to the frame { OX θ , OY θ }). The results obtained represent a general solution of M. de Guzmán's problem that was previously studied by various authors.

We consider the quantifier-free set theory MLSUn containing the symbols U , \, =, ∈, Un . Un ( p ) is interpreted as the union of all members of the set p . It is proved that there exists an algorithm which for any formula Q of the MLSUn theory containing at most one occurrence of the symbol Un decides whether Q is true or not using the space cn 3 log 2 n ( n is the length of Q ).