The paper presents a study of axiomatic theory of quadratic forms. Two operations on quadratic form schemes are investigated: the product of schemes and the group extension of schemes. The main result states that the product of schemes realized by fields is again realized by a field.

Let g be an elementary 2-group with -1∈g and let d be a mapping of g into the family of all subgroups of g. The triple S = (g,-1,d) is called a quadratic form scheme if C1-C3 are fulfilled. The main result is: if |g| ≤ 16 or [g : R] ≤ 16 then all these schemes can be obtained as the product of schemes or schemes of form S^t from the schemes of fields C, R, F₃, F₅, Q₂(√-1), Q₂(√-2) and radical schemes S_i^β. We give a complete list of schemes for |g| ≤ 16, [g : R] ≤ 16 with all invariants.

In this paper the existence and uniqueness of stochastic solutions of equation (1) are presented. Our theorem is an application of Banach fixed point theorem to the integral equation with advancing argument.

This paper considers the various types of controllability of linear infinite-dimensional dynamical systems defined in a Banach space, with multiple time-varying delays in control. Necessary and sufficient conditions for approximate controllability, approximate relative controllability and approximate absolute controllability of these systems are obtained. Special cases of systems defined in a Hilbert space are also considered.

The paper deals with the problem of the behaviour of a given solution of a quasi-linear parabolic equation near the parabolic boundary. Necessary and sufficient conditions for weak and strong convergence in the Sobolev space W_p^1,1, p ≥ 2, are given.

In this paper the differential-functional equation
ϕ'(t) = f(t,ϕ), t∈R⁺
ϕ(0) = η
with an unbounded advanced argument is discovered. Under suitable assumptions, using Schauder Fixed Point Principle, a theorem on existence of solutions in a special functional-space is proved.

In the present paper we deal with local analytic solutions of the functional equation announced in the title. This paper is a continuation of [2] and [3].

In the present paper we consider the problem of the existence and uniqueness of local analytic solutions of the equation ϕ(f(x)) = V(ϕ(x))+U(x). This paper is a continuation of [1], [2] and [3].

We shall deal with the existence and dependence on an arbitrary function of solutions of the functional equation
(1)     ϕ(f(x)) = g(x,ϕ(x))
in the class of functions fulfilling a Lipschitz condition.

The paper deals with the existence and uniqueness of regular solutions of the equation ϕ(x) = h(x,ϕ(f(x))). Also in the indeterminate case the existence of solutions of ϕ(f(x)) = g(x,ϕ(x)) is studied.

By a (general) convergence in a given linear space X we mean a mapping G: X^N→2^X, where N denotes the set of all positive integers, and by a zero-convergence in X we mean a convergence G₀ in X for which G₀(x)≠⌀ implies 0∈G₀(x) for each x∈X^N. In the paper, the two operations are defined: 1° operation C, which to each zero-convergence G₀ in X assigns some general convergence G in X, and 2° operation C₀, which to each general convergence G in X assigns a zero-convergence G₀ in X. Various systems of axioms for general convergences and zero-convergences are considered and their connections with the operations C and C₀ are studied. Also mutual independence of axioms is studied.

Certain properties of solutions of ordinary linear differential equations have an algebraic character. In this note we are concerned with these properties. Our purpose is to show some connections between the theory of differential rings and the theory of linear spaces equipped with an endomorphism satisfying some additional conditions. It will be shown that some linear spaces may be extended to differential rings.

The first part of the paper is devoted to topological analogies of some theorems from real analysis (Vitali’s covering theorem, Lebesgue theorem on outer density, Smital’s lemma). Then we give a topological analogue of a theorem of Ostrowski connected with additive functions.
In the last part we deal with Hamel bases, specially with Burstin bases. We prove their existence and study topological and measure properties.

Two relations between families of pseudonorms are considered: relation of being weaker than and relation of being sequentially weaker than. Connections between these concepts are discussed.

In this note we give the form of generators and co-generators of semigroups of „substitution operators” in Banach space C([a,b]). We also establish some properties of these operators related to Schröder equation.