Let R be a semi-local ring with 2∈U(R) and such that all residue class fields of R contain more than 3 elements. It is proved here that bilinear spaces over R are classified by dimension, determinant, Hasse invariant and total signature if and only if the third power of the fundamental ideal of Witt ring W(R) is torsion free. This is a generalization of the same result when R is a field due to Elman and Lam.

Let k be any field of characteristic different from 2, F will denote the ring of formal power series in two variables with coefficients from k and K its field of quotients. The aim of the paper is to investigate the structure of the Witt ring of K— W(K). We shall construct certain exact and split sequences of additive homomorphism. We count special the cases of complex and real
number fields. The results are also valid for rings of Nash series.

Let C be an M-family of subsets of X and C₁ - the family of its “first category” sets. It is proven that one and only one of the following conditions is satisfied: (*) each C₁-set is at most countable; (**) X is the union of C₁ set and a set having property (L), which are disjoint; (***) each C-residual set contains an uncountable C₁-set.
Moreover, if C⊂2^X and D⊂2^Y are two M-families, the “duality principle” holds (i.e. there exists a bijection f: X→Y transforming C₁-sets onto D₁-sets) iff C and D satisfy the same of the conditions above.
Also, some considerations are added, concerning the coincidence between the properties of the family C₁ and a σ-ideal.

Let R be the set of all real numbers. In the present paper we shall characterize functions f: R→R which are either linear or have graph contained and dense in the plane or half-plane determined by a linear function. For this purpose we consider functions satisfying certain limitary conditions which are related to the additivity equation but considerably weaker than that.

In this paper we consider a set-valued function of two variables, measurable in the first and continuous in the second variable. Using metric projections we construct for this function a family of selectors which are Carathéodory maps. The existence of Carathéodory selectors was studied by Castaing [2], [3], Cellina [4], Fryszkowski [9] and the first author [11].

Recently several authors demonstrated random fixed point theorems for various classes of multifunctions ([7], [8], [2], [3], [12], [10]). On the other hand we do not know any work on applications of these theorems. In this paper we apply to games and dynamic programming a random analogue of the Fan-Kakutani fixed point theorem. We consider a zero-sum two-person game depending on a random parameter, and present sufficient conditions for the existence of a measurable solution. Then we study the existence of measurable stationary optimal programs in discounted dynamic programming with a random parameter.

The literature devoted to differential-functional equations with advancing argument is rather scarce. In the papers [1], [2], [4], [7] the existence of solutions of Cauchy problem and in [5], [6] and [8] - of Nicoletti problem for differential-functional equations with bounded advancement of argument are investigated. The differential-functional equations with unbounded advancement of argument are considered in the articles [3] and [9]. Namely in [3] the existence and uniqueness of solution of the Nicoletti problem is proved, and in [9] an existence theorem for the Cauchy problem is given. The purpose of this paper is to formulate an existence theorem for the Nicoletti problem in the case where the advancement of the argument is unbounded. The proof of this theorem is based on Schauder’s fixed point theorem.

In this work we introduce new types of convolution products on the set of the complex-valued continuous functions defined in [0,∞] and we see that there is a geometrical interpretation. Then we show that the corresponding fields of fractions are isomorphic to the classical field of the Mikusiński operators. The idea of transport of structure is essential in this work.

In this paper we investigate some properties of solutions of the heat equation. Their basic properties are established in [3]. Our object is to prove some partial distribution function inequalities for the area integral which can be used to study the local and the global behavior of solutions of the heat equation. Theorem 3 shows that the area integral A and the nontangential maximal function N are remarkably closely related. The method used in this paper is based on the treatment of analogous problems for harmonic functions in [1].

We give a simple proof that, under CH, βN-N-{p} is not normal for any p∈βN-N.

The paper is devoted to inequalities between π₀(X) and π_d(X) where
π₀(X) := min{π(U): U open and non-empty subset of X},
π_d(X) := min{|B|: every open and dense subset of X containes an element from B}.
From these definitions π_d(X) ≤ π₀(X) for every space X. In the paper we construct a space X for which π_d(X)=ω₁ and π₀(X)=2^ℵ₀.

In this note we construct maps between metric separable connected spaces X and Y such that the graphs are connected, dense and rigid subspaces of the Cartesian product X×Y. From this result it follows that there is no maximal topology among metric separable connected topologies on a given set X.

The paper contains a construction of a Tychonoff space X such that for every compact extension bX the subset bX-X contains a non-empty S_δ-set G such that Int G = ⌀.

In the New Scottish Book M. Katĕtov asked whether there exists a Hausdorff space X without isolated points such that every real-valued function on X is continuous at some point? In the paper it is shown that the existence of such a space is equiconsistent to the existence of measurable cardinal.

We give conditions under which iterated hyperspaces of finite subsets, with Ochan’s topology, are homeomorphic.