Let F and O_F be a number field and its ring of integers respectively. Let K₂ denote Milnor K-functor. In the paper we describe the structure of the group K₂O_F/R₂F, where R₂F is the Hilbert kernel and F is multiquadratic extension of the rational number field. Moreover, we give some characterization of fields with trivial group K₂O_F/R₂F. At the end we make some remarks on p-rank of K₂O_F and divisibility of the ideal class group by 2.

In the paper the number c(S) of closed subgroups of a quadratic form scheme S is considered. We determine c(S) for some classes of schemes. Schemes with small c(S) are characterized and the behaviour of c(S) under known operation on schemes are discussed, as well.

We show that the generally accepted axioms C1, C2 and C3 of quadratic form schemes can be replaced by two simply axioms S1 and S2. Several operations on schemes and the notion of a complete scheme are introduced in the paper. A proof is given that every elementary scheme is complete.

Fundamental properties of equations of the form (1) are discussed from the Baire category point of view.  After showing that they are generic in a suitable function space the density of the set of equations (1) having no solutions is studied. Results of the paper are “product versions” of these proved in [3].

Consider the sequence of inequalities
a_i,k+1 ≤ Γ(Σ_j=1ⁿs_i,j(a_1,k,...,a_n,k)) + b_i,k,
i = 1,2,...,n, k = 0,1,2,..., where Γ is a concave and increasing function and the functions s_i,j, i,j = 1,...,n, satisfy some additional conditions ((2.2) and (2.3)). In this note we give an answer to the following question: under what assumptions regarding the given sequences {b_i,k} their behaviour mplies a similar one for the sequences {a_i,k}.

The author shows how to fill a gap occurring in [3; § 13.1].

On examine en quel degré les suppositions dans un théorème de S. Gołąb, au sujet de la détermination au juste de 1’isomorphisme de 1’addition et de la multiplication simples, sont essentielles. On démontre aussi une modification de ce théorème et on pose un problème ouvert.

In this paper we prove that every measurable quadratic stochastic process X : R^N×Ω→R is continuous and has the form
X(x,·) = Σ_i,j=1^Nx_ix_jY_i,j(·)     (a.e.),
where x = (x₁,...,x_N)∈R^N and Y_i,j: Ω→R are random variables. Moreover, we give a proof of the stability of the quadratic stochastic processes.

A characterization of the exponential distribution via a functional equation on a restricted domain is given.

We prove theorem on the existence and uniqueness of the distributional solutions of the Cauchy problem for equation (1.0).

In this paper we prove a theorem on the existence and uniqueness of the solution of the Cauchy problem for linear differential equations with distributional coefficients.

The paper refers to the subject of the estimation of the difference between the solutions of two delated stochastic equations, which as a consequence gives also some criterions of the uniqueness for these equations. The results are obtained by using some integral inequalities and applying them to the more general class of equations with local integrable martingales.

The paper contains a generalization of some uniqueness criterions for the Itô’s differential equations established by Skorokhod, Yamada and Watanabe. The results were generalized by applying more general non-linear integral inequalities and hence the stochastic versions of uniqueness criterions for non-random differential equations were obtained.

The main result of the paper is the estimation of the operational function of the exponential type which is the parametric one in the sense of the Mikusinski’s theory. The estimation may be applied in studying the behaviour of solutions of partial differential equations.

In this paper we consider the problem of optimal control for general stochastic differential equation of Itô type. We prove the existence of solutions of this equation under weaker assumptions than in [2]. Moreover, we prove the compactness of the space of solutions and the existence of optimal control.

The algebraical and topological structure of the set S? (resp. ????₁) of all n×m matrix games with saddle points (resp. with unique saddle point) in the space R^n×m of all such games is studied. It has been shown that ???? is a closed cone with vertex zero and includes the origin. Moreover, it is neither convex nor dense subset of R^n×m. The set ????₁ is a non-convex cone which does not include the origin. It is neither closed nor open.
The concept of “reserve of non-saddlexity” has been also introduced.

It has been proved, by K. Menger in [3], that all concepts of the Bolyai-Lobachevsky geometry can be defined in terms of the operations “joining” and “intersecting”. In the present paper a similar definition of the orthogonality on hyperbolic plane (being a substitute for the Bolyai-Lobachevsky plane) over a finite field with characteristic different from 2 or over a finite extension of the rational field to a subfield of the real field is given and investigated.