In the paper it is shown that the Witt group of the rational function field in countably many variables over a real-closed field can be decomposed into direct sum of cyclic groups. This is an example that the sufficient conditions given in [3] are not necessary.

We prove that the Yucas quaternionic mapping (considered in [6]) satisfies B_G = X₃(a) for any a∈B_G. We also give examples of a quaternionic mapping which satisfies B_G = X₂(a) for any a∈B_G but not (L) and a quaternionic mapping which satisfies B_G = X₁(a) for any a∈B_G but not (L).

In this paper, we evaluate two integrals involving Hermite polynomial and confluent hypergeometric function, and employ one of them to evaluate an integral involving Meijer's G-function. We use the integral involving Meijer's G-function to evaluate a double integral involving Meijer's G-function. We further employ the integral involving Meijer's G-function to establish one one-dimensional Fourier-Hermite expansion and one two-dimensional Fourier-Hermite expansion. We also obtain a solution of a heat conduction problem.

Let l^ϕ be a Musielak-Orlicz sequence space. Let X¹_ϕ and X_ϕ be the modular spaces of multifunctions generated by l^ϕ. Let K_w,j: R→R for j = 0,1,2,..., w∈W, where W is an abstract set of indices. Assuming certain singularity assumption on the nonlinear kernel K_w,j and setting T_w(F)=(T_w(F)(i))_i=0^∞ with (T_w(F))(i) = {Σ_j=0ⁱK_w,j(f(j)) : f(j)∈F(j)}, convergence theorems T_w(F)→_ϕ F in X¹_ϕ and T_w(F)→_d,ϕ, F in X_ϕ are obtained.

There is defined quasi-Jensen function as a solution of a certain functional inequality which generalizes the classical Jensen equation: f((x+y)/2) = (f(x)+f(y))/2. The introduced inequality is analogous to the inequality which defines J. Tabor's quasi-additive functions. The main result of this paper is to show strong relationship between quasi-Jensen and quasi-additive functions.

It is shown (under suitable conditions on H⊂R) that if f: R→R is a measurable function such that for an n∈N₀ and every h∈H we have Δhⁿ⁺¹f(x) = 0 almost everywhere on R, then f is equal almost everywhere on R to a polynomial of degree at most n. In particular, every measurable polynomial function f: R→R is a polynomial. In fact, these (essentially known) results are here proved in a more general and more abstract form. The paper contains also a version of the Łomnicki-type theorem on measurable microperiodic functions.

In this note we show that there exists a collection containing c additive functions with big graphs such that f(x) ≠ g{x) for every f,g in the collection (f ≠ g) and every x∈Rⁿ\{0}.

In this paper a certain natural generalization of bilinear functional is introduced and investigated. We define quasibilinear functionals by replacing the additivity of a bilinear functional with three weaker conditions. The solution is a sequence of bilinear functionals on subspaces of the given linear space.

In the preceding paper (see [2]) we defined and investigated quasibilinear functionals on vector spaces, quasiorthogonal and weakly quasiorthogonal vector spaces. In the present paper we give certain applications of these concepts in projective geometry.

In [1], using the notion of linear space of translations of the set over the field, n-dimensional Klein spaces over arbitrary field were defined. In [2] the definition of vector structure over the field was given and used to introduce the concept of n-dimensional generalized elementary Klein space.
The aim of present paper is to define (Section 1) and state some of the properties of, so called, spaces with vector structure, without the use of Klein's ideas. In Section 2 it is shown that afiine and Euclidean space are the examples of such spaces. Other examples are the elliptic and projective space. Using the notion of vector structure, in Section 3 the definition of tangent bundle is given and some properties of it are observed, with the aim to introduce (Section 4) the concept of m-dimensional hyperplane in spaces with vector structure.